cp-algo/number_theory/two_squares.hpp

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• Last update: 2025-12-17 02:23:31+01:00
• Include: #include "cp-algo/number_theory/two_squares.hpp"

Depends on

• cp-algo/math/common.hpp
• cp-algo/number_theory/euler.hpp
• cp-algo/number_theory/factorize.hpp
• cp-algo/number_theory/modint.hpp
• cp-algo/number_theory/primality.hpp
• cp-algo/random/rng.hpp
• cp-algo/util/big_alloc.hpp
• cp-algo/util/complex.hpp
• cp-algo/util/simd.hpp

Required by

• cp-algo/number_theory.hpp

Verified with

• Represent A Number As Two Square Sum (verify/number_theory/two_squares.test.cpp)

Code

#ifndef CP_ALGO_NUMBER_THEORY_TWO_SQUARES_HPP
#define CP_ALGO_NUMBER_THEORY_TWO_SQUARES_HPP
#include "euler.hpp"
#include "../util/complex.hpp"
#include "../util/big_alloc.hpp"
#include <cassert>
#include <utility>
#include <vector>
#include <map>
namespace cp_algo::math {
    template<typename T>
    using gaussint = complex<T>;
    template<typename _Int>
    auto two_squares_prime_any(_Int p) {
        if(p == 2) {
            return gaussint<_Int>{1, 1};
        }
        assert(p % 4 == 1);
        using Int = std::make_signed_t<_Int>;
        using base = dynamic_modint<Int>;
        return base::with_mod(p, [&](){
            base g = primitive_root(p);
            int64_t i = bpow(g, (p - 1) / 4).getr();
            int64_t q0 = 1, q1 = 0;
            int64_t r = i, m = p;
            // TODO: Use library contfrac?
            do {
                int64_t d = r / m;
                q0 = std::exchange(q1, q0 + d * q1);
                r = std::exchange(m, r % m);
            } while(q1 < p / q1);
            return gaussint<_Int>{q0, (base(i) * base(q0)).rem()};
        });
    }

    template<typename Int>
    big_vector<gaussint<Int>> two_squares_all(Int n) {
        if(n == 0) {
            return {0};
        }
        auto primes = factorize(n);
        big_map<Int, int> cnt;
        for(auto p: primes) {
            cnt[p]++;
        }
        big_vector<gaussint<Int>> res = {1};
        for(auto [p, c]: cnt) {
            big_vector<gaussint<Int>> nres;
            if(p % 4 == 3) {
                if(c % 2 == 0) {
                    auto mul = bpow(gaussint<Int>(p), c / 2);
                    for(auto p: res) {
                        nres.push_back(p * mul);
                    }
                }
            } else if(p % 4 == 1) {
                auto base = two_squares_prime_any(p);
                for(int i = 0; i <= c; i++) {
                    auto mul = bpow(base, i) * bpow(conj(base), c - i);
                    for(auto p: res) {
                        nres.push_back(p * mul);
                    }
                }
            } else if(p % 4 == 2) {
                auto mul = bpow(gaussint<Int>(1, 1), c);
                for(auto p: res) {
                    nres.push_back(p * mul);
                }
            }
            res = nres;
        }
        big_vector<gaussint<Int>> nres;
        for(auto p: res) {
            while(p.real() < 0 || p.imag() < 0) {
                p *= gaussint<Int>(0, 1);
            }
            nres.push_back(p);
            if(!p.real() || !p.imag()) {
                nres.emplace_back(p.imag(), p.real());
            }
        }
        return nres;
    }
}
#endif // CP_ALGO_NUMBER_THEORY_TWO_SQUARES_HPP

#line 1 "cp-algo/number_theory/two_squares.hpp"


#line 1 "cp-algo/number_theory/euler.hpp"


#line 1 "cp-algo/number_theory/factorize.hpp"


#line 1 "cp-algo/number_theory/primality.hpp"


#line 1 "cp-algo/number_theory/modint.hpp"


#line 1 "cp-algo/math/common.hpp"


#include <functional>
#include <cstdint>
#include <cassert>
namespace cp_algo::math {
#ifdef CP_ALGO_MAXN
    const int maxn = CP_ALGO_MAXN;
#else
    const int maxn = 1 << 19;
#endif
    const int magic = 64; // threshold for sizes to run the naive algo

    auto bpow(auto const& x, auto n, auto const& one, auto op) {
        if(n == 0) {
            return one;
        } else {
            auto t = bpow(x, n / 2, one, op);
            t = op(t, t);
            if(n % 2) {
                t = op(t, x);
            }
            return t;
        }
    }
    auto bpow(auto x, auto n, auto ans) {
        return bpow(x, n, ans, std::multiplies{});
    }
    template<typename T>
    T bpow(T const& x, auto n) {
        return bpow(x, n, T(1));
    }
    inline constexpr auto inv2(auto x) {
        assert(x % 2);
        std::make_unsigned_t<decltype(x)> y = 1;
        while(y * x != 1) {
            y *= 2 - x * y;
        }
        return y;
    }
}

#line 4 "cp-algo/number_theory/modint.hpp"
#include <iostream>
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math {

    template<typename modint, typename _Int>
    struct modint_base {
        using Int = _Int;
        using UInt = std::make_unsigned_t<Int>;
        static constexpr size_t bits = sizeof(Int) * 8;
        using Int2 = std::conditional_t<bits <= 32, int64_t, __int128_t>;
        using UInt2 = std::conditional_t<bits <= 32, uint64_t, __uint128_t>;
        constexpr static Int mod() {
            return modint::mod();
        }
        constexpr static Int remod() {
            return modint::remod();
        }
        constexpr static UInt2 modmod() {
            return UInt2(mod()) * mod();
        }
        constexpr modint_base() = default;
        constexpr modint_base(Int2 rr) {
            to_modint().setr(UInt((rr + modmod()) % mod()));
        }
        modint inv() const {
            return bpow(to_modint(), mod() - 2);
        }
        modint operator - () const {
            modint neg;
            neg.r = std::min(-r, remod() - r);
            return neg;
        }
        modint& operator /= (const modint &t) {
            return to_modint() *= t.inv();
        }
        modint& operator *= (const modint &t) {
            r = UInt(UInt2(r) * t.r % mod());
            return to_modint();
        }
        modint& operator += (const modint &t) {
            r += t.r; r = std::min(r, r - remod());
            return to_modint();
        }
        modint& operator -= (const modint &t) {
            r -= t.r; r = std::min(r, r + remod());
            return to_modint();
        }
        modint operator + (const modint &t) const {return modint(to_modint()) += t;}
        modint operator - (const modint &t) const {return modint(to_modint()) -= t;}
        modint operator * (const modint &t) const {return modint(to_modint()) *= t;}
        modint operator / (const modint &t) const {return modint(to_modint()) /= t;}
        // Why <=> doesn't work?..
        auto operator == (const modint &t) const {return to_modint().getr() == t.getr();}
        auto operator != (const modint &t) const {return to_modint().getr() != t.getr();}
        auto operator <= (const modint &t) const {return to_modint().getr() <= t.getr();}
        auto operator >= (const modint &t) const {return to_modint().getr() >= t.getr();}
        auto operator < (const modint &t) const {return to_modint().getr() < t.getr();}
        auto operator > (const modint &t) const {return to_modint().getr() > t.getr();}
        Int rem() const {
            UInt R = to_modint().getr();
            return R - (R > (UInt)mod() / 2) * mod();
        }
        constexpr void setr(UInt rr) {
            r = rr;
        }
        constexpr UInt getr() const {
            return r;
        }

        // Only use these if you really know what you're doing!
        static UInt modmod8() {return UInt(8 * modmod());}
        void add_unsafe(UInt t) {r += t;}
        void pseudonormalize() {r = std::min(r, r - modmod8());}
        modint const& normalize() {
            if(r >= (UInt)mod()) {
                r %= mod();
            }
            return to_modint();
        }
        void setr_direct(UInt rr) {r = rr;}
        UInt getr_direct() const {return r;}
    protected:
        UInt r;
    private:
        constexpr modint& to_modint() {return static_cast<modint&>(*this);}
        constexpr modint const& to_modint() const {return static_cast<modint const&>(*this);}
    };
    template<typename modint>
    concept modint_type = std::is_base_of_v<modint_base<modint, typename modint::Int>, modint>;
    template<modint_type modint>
    decltype(std::cin)& operator >> (decltype(std::cin) &in, modint &x) {
        typename modint::UInt r;
        auto &res = in >> r;
        x.setr(r);
        return res;
    }
    template<modint_type modint>
    decltype(std::cout)& operator << (decltype(std::cout) &out, modint const& x) {
        return out << x.getr();
    }

    template<auto m>
    struct modint: modint_base<modint<m>, decltype(m)> {
        using Base = modint_base<modint<m>, decltype(m)>;
        using Base::Base;
        static constexpr Base::Int mod() {return m;}
        static constexpr Base::UInt remod() {return m;}
        auto getr() const {return Base::r;}
    };

    template<typename Int = int>
    struct dynamic_modint: modint_base<dynamic_modint<Int>, Int> {
        using Base = modint_base<dynamic_modint<Int>, Int>;
        using Base::Base;

        static Base::UInt m_reduce(Base::UInt2 ab) {
            if(mod() % 2 == 0) [[unlikely]] {
                return typename Base::UInt(ab % mod());
            } else {
                typename Base::UInt2 m = typename Base::UInt(ab) * imod();
                return typename Base::UInt((ab + m * mod()) >> Base::bits);
            }
        }
        static Base::UInt m_transform(Base::UInt a) {
            if(mod() % 2 == 0) [[unlikely]] {
                return a;
            } else {
                return m_reduce(a * pw128());
            }
        }
        dynamic_modint& operator *= (const dynamic_modint &t) {
            Base::r = m_reduce(typename Base::UInt2(Base::r) * t.r);
            return *this;
        }
        void setr(Base::UInt rr) {
            Base::r = m_transform(rr);
        }
        Base::UInt getr() const {
            typename Base::UInt res = m_reduce(Base::r);
            return std::min(res, res - mod());
        }
        static Int mod() {return m;}
        static Int remod() {return 2 * m;}
        static Base::UInt imod() {return im;}
        static Base::UInt2 pw128() {return r2;}
        static void switch_mod(Int nm) {
            m = nm;
            im = m % 2 ? inv2(-m) : 0;
            r2 = static_cast<Base::UInt>(static_cast<Base::UInt2>(-1) % m + 1);
        }

        // Wrapper for temp switching
        auto static with_mod(Int tmp, auto callback) {
            struct scoped {
                Int prev = mod();
                ~scoped() {switch_mod(prev);}
            } _;
            switch_mod(tmp);
            return callback();
        }
    private:
        static thread_local Int m;
        static thread_local Base::UInt im, r2;
    };
    template<typename Int>
    Int thread_local dynamic_modint<Int>::m = 1;
    template<typename Int>
    dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::im = -1;
    template<typename Int>
    dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::r2 = 0;
}

#line 4 "cp-algo/number_theory/primality.hpp"
#include <algorithm>
#include <bit>
namespace cp_algo::math {
    // https://en.wikipedia.org/wiki/Miller–Rabin_primality_test
    template<typename _Int>
    bool is_prime(_Int m) {
        using Int = std::make_signed_t<_Int>;
        using UInt = std::make_unsigned_t<Int>;
        if(m == 1 || m % 2 == 0) {
            return m == 2;
        }
        // m - 1 = 2^s * d
        int s = std::countr_zero(UInt(m - 1));
        auto d = (m - 1) >> s;
        using base = dynamic_modint<Int>;
        auto test = [&](base x) {
            x = bpow(x, d);
            if(std::abs(x.rem()) <= 1) {
                return true;
            }
            for(int i = 1; i < s && x != -1; i++) {
                x *= x;
            }
            return x == -1;
        };
        return base::with_mod(m, [&]() {
#ifdef CP_ALGO_NUMBER_THEORY_PRIMALITY_BASES_HPP
            uint16_t base2 = 7, base3 = 61;
            if (m != uint32_t(m)) {
                base2 = base_table1[uint32_t(m * 0xAD625B89) >> 18];
                base3 = base_table2[base2 >> 13];
            }
            return test(2) && test(base2) && test(base3);
#else
            return std::ranges::all_of(std::array{2, 325, 9375, 28178, 450775, 9780504, 1795265022}, test);
#endif
        });
    }
}

#line 1 "cp-algo/util/big_alloc.hpp"



#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#include <vector>
#include <string>
#include <cstddef>
#line 13 "cp-algo/util/big_alloc.hpp"
#include <generator>
#include <forward_list>

// Single macro to detect POSIX platforms (Linux, Unix, macOS)
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
#  define CP_ALGO_USE_MMAP 1
#  include <sys/mman.h>
#else
#  define CP_ALGO_USE_MMAP 0
#endif

namespace cp_algo {
    template <typename T, size_t Align = 32>
    class big_alloc {
        static_assert( Align >= alignof(void*), "Align must be at least pointer-size");
        static_assert(std::popcount(Align) == 1, "Align must be a power of two");
    public:
        using value_type = T;
        template <class U> struct rebind { using other = big_alloc<U, Align>; };
        constexpr bool operator==(const big_alloc&) const = default;
        constexpr bool operator!=(const big_alloc&) const = default;

        big_alloc() noexcept = default;
        template <typename U, std::size_t A>
        big_alloc(const big_alloc<U, A>&) noexcept {}

        [[nodiscard]] T* allocate(std::size_t n) {
            std::size_t padded = round_up(n * sizeof(T));
            std::size_t align = std::max<std::size_t>(alignof(T),  Align);
#if CP_ALGO_USE_MMAP
            if (padded >= MEGABYTE) {
                void* raw = mmap(nullptr, padded,
                                PROT_READ | PROT_WRITE,
                                MAP_PRIVATE | MAP_ANONYMOUS, -1, 0);
                madvise(raw, padded, MADV_HUGEPAGE);
                madvise(raw, padded, MADV_POPULATE_WRITE);
                return static_cast<T*>(raw);
            }
#endif
            return static_cast<T*>(::operator new(padded, std::align_val_t(align)));
        }

        void deallocate(T* p, std::size_t n) noexcept {
            if (!p) return;
            std::size_t padded = round_up(n * sizeof(T));
            std::size_t align  = std::max<std::size_t>(alignof(T),  Align);
    #if CP_ALGO_USE_MMAP
            if (padded >= MEGABYTE) { munmap(p, padded); return; }
    #endif
            ::operator delete(p, padded, std::align_val_t(align));
        }

    private:
        static constexpr std::size_t MEGABYTE = 1 << 20;
        static constexpr std::size_t round_up(std::size_t x) noexcept {
            return (x + Align - 1) / Align * Align;
        }
    };

    template<typename T> using big_vector = std::vector<T, big_alloc<T>>;
    template<typename T> using big_basic_string = std::basic_string<T, std::char_traits<T>, big_alloc<T>>;
    template<typename T> using big_deque = std::deque<T, big_alloc<T>>;
    template<typename T> using big_stack = std::stack<T, big_deque<T>>;
    template<typename T> using big_queue = std::queue<T, big_deque<T>>;
    template<typename T> using big_priority_queue = std::priority_queue<T, big_vector<T>>;
    template<typename T> using big_forward_list = std::forward_list<T, big_alloc<T>>;
    using big_string = big_basic_string<char>;

    template<typename Key, typename Value, typename Compare = std::less<Key>>
    using big_map = std::map<Key, Value, Compare, big_alloc<std::pair<const Key, Value>>>;
    template<typename T, typename Compare = std::less<T>>
    using big_multiset = std::multiset<T, Compare, big_alloc<T>>;
    template<typename T, typename Compare = std::less<T>>
    using big_set = std::set<T, Compare, big_alloc<T>>;
    template<typename Ref, typename V = void>

    using big_generator = std::generator<Ref, V, big_alloc<std::byte>>;
}

// Deduction guide to make elements_of with big_generator default to big_alloc
namespace std::ranges {
    template<typename Ref, typename V>
    elements_of(cp_algo::big_generator<Ref, V>&&) -> elements_of<cp_algo::big_generator<Ref, V>&&, cp_algo::big_alloc<std::byte>>;
}


#line 1 "cp-algo/random/rng.hpp"


#include <chrono>
#include <random>
namespace cp_algo::random {
    std::mt19937_64 gen(
        std::chrono::steady_clock::now().time_since_epoch().count()
    );
    uint64_t rng() {
        return gen();
    }
}

#line 7 "cp-algo/number_theory/factorize.hpp"
namespace cp_algo::math {
    // https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm
    template<typename _Int>
    auto proper_divisor(_Int m) {
        using Int = std::make_signed_t<_Int>;
        using base = dynamic_modint<Int>;
        return m % 2 == 0 ? 2 : base::with_mod(m, [&]() {
            base t = random::rng();
            auto f = [&](auto x) {
                return x * x + t;
            };
            base x = 0, y = 0;
            base g = 1;
            while(g == 1) {
                for(int i = 0; i < 64; i++) {
                    x = f(x);
                    y = f(f(y));
                    if(x == y) [[unlikely]] {
                        t = random::rng();
                        x = y = 0;
                    } else {
                        base t = g * (x - y);
                        g = t == 0 ? g : t;
                    }
                }
                g = std::gcd(g.getr(), m);
            }
            return g.getr();
        });
    }
    template<typename Int>
    big_generator<Int> factorize(Int m) {
        if(is_prime(m)) {
            co_yield m;
        } else if(m > 1) {
            auto g = proper_divisor(m);
            co_yield std::ranges::elements_of(factorize(g));
            co_yield std::ranges::elements_of(factorize(m / g));
        }
    }
    template<typename Int>
    big_generator<Int> divisors_sqrt(Int m) {
        for(Int i = 1; i * i <= m; i++) {
            if(m % i == 0) {
                co_yield i;
                if(i * i != m) {
                    co_yield m / i;
                }
            }
        }
    }
}

#line 5 "cp-algo/number_theory/euler.hpp"
namespace cp_algo::math {
    auto euler_phi(auto m) {
        using T = std::decay_t<decltype(m)>;
        auto primes = to<big_vector<T>>(factorize(m));
        std::ranges::sort(primes);
        auto [from, to] = std::ranges::unique(primes);
        primes.erase(from, to);
        auto ans = m;
        for(auto it: primes) {
            ans -= ans / it;
        }
        return ans;
    }
    template<modint_type base>
    auto period(base x) {
        auto ans = euler_phi(base::mod());
        base x0 = bpow(x, ans);
        for(auto t: factorize(ans)) {
            while(ans % t == 0 && x0 * bpow(x, ans / t) == x0) {
                ans /= t;
            }
        }
        return ans;
    }
    template<typename _Int>
    _Int primitive_root(_Int p) {
        using Int = std::make_signed_t<_Int>;
        using base = dynamic_modint<Int>;
        return base::with_mod(p, [p](){
            base t = 1;
            while(period(t) != p - 1) {
                t = random::rng();
            }
            return t.getr();
        });
    }
}

#line 1 "cp-algo/util/complex.hpp"


#line 4 "cp-algo/util/complex.hpp"
#include <cmath>
#include <type_traits>
#line 1 "cp-algo/util/simd.hpp"


#include <experimental/simd>
#line 6 "cp-algo/util/simd.hpp"
#include <memory>

#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_SIMD_AVX2_TARGET _Pragma("GCC target(\"avx2\")")
#else
#define CP_ALGO_SIMD_AVX2_TARGET
#endif

#define CP_ALGO_SIMD_PRAGMA_PUSH \
    _Pragma("GCC push_options") \
    CP_ALGO_SIMD_AVX2_TARGET

CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo {
    template<typename T, size_t len>
    using simd [[gnu::vector_size(len * sizeof(T))]] = T;
    using u64x8 = simd<uint64_t, 8>;
    using u32x16 = simd<uint32_t, 16>;
    using i64x4 = simd<int64_t, 4>;
    using u64x4 = simd<uint64_t, 4>;
    using u32x8 = simd<uint32_t, 8>;
    using u16x16 = simd<uint16_t, 16>;
    using i32x4 = simd<int32_t, 4>;
    using u32x4 = simd<uint32_t, 4>;
    using u16x8 = simd<uint16_t, 8>;
    using u16x4 = simd<uint16_t, 4>;
    using i16x4 = simd<int16_t, 4>;
    using u8x32 = simd<uint8_t, 32>;
    using u8x8 = simd<uint8_t, 8>;
    using u8x4 = simd<uint8_t, 4>;
    using dx4 = simd<double, 4>;

    inline dx4 abs(dx4 a) {
        return dx4{
            std::abs(a[0]),
            std::abs(a[1]),
            std::abs(a[2]),
            std::abs(a[3])
        };
    }

    // https://stackoverflow.com/a/77376595
    // works for ints in (-2^51, 2^51)
    static constexpr dx4 magic = dx4() + (3ULL << 51);
    inline i64x4 lround(dx4 x) {
        return i64x4(x + magic) - i64x4(magic);
    }
    inline dx4 to_double(i64x4 x) {
        return dx4(x + i64x4(magic)) - magic;
    }

    inline dx4 round(dx4 a) {
        return dx4{
            std::nearbyint(a[0]),
            std::nearbyint(a[1]),
            std::nearbyint(a[2]),
            std::nearbyint(a[3])
        };
    }

    inline u64x4 low32(u64x4 x) {
        return x & uint32_t(-1);
    }
    inline auto swap_bytes(auto x) {
        return decltype(x)(__builtin_shufflevector(u32x8(x), u32x8(x), 1, 0, 3, 2, 5, 4, 7, 6));
    }
    inline u64x4 montgomery_reduce(u64x4 x, uint32_t mod, uint32_t imod) {
#ifdef __AVX2__
        auto x_ninv = u64x4(_mm256_mul_epu32(__m256i(x), __m256i() + imod));
        x += u64x4(_mm256_mul_epu32(__m256i(x_ninv), __m256i() + mod));
#else
        auto x_ninv = u64x4(u32x8(low32(x)) * imod);
        x += x_ninv * uint64_t(mod);
#endif
        return swap_bytes(x);
    }

    inline u64x4 montgomery_mul(u64x4 x, u64x4 y, uint32_t mod, uint32_t imod) {
#ifdef __AVX2__
        return montgomery_reduce(u64x4(_mm256_mul_epu32(__m256i(x), __m256i(y))), mod, imod);
#else
        return montgomery_reduce(x * y, mod, imod);
#endif
    }
    inline u32x8 montgomery_mul(u32x8 x, u32x8 y, uint32_t mod, uint32_t imod) {
        return u32x8(montgomery_mul(u64x4(x), u64x4(y), mod, imod)) |
               u32x8(swap_bytes(montgomery_mul(u64x4(swap_bytes(x)), u64x4(swap_bytes(y)), mod, imod)));
    }
    inline dx4 rotate_right(dx4 x) {
        static constexpr u64x4 shuffler = {3, 0, 1, 2};
        return __builtin_shuffle(x, shuffler);
    }

    template<std::size_t Align = 32>
    inline bool is_aligned(const auto* p) noexcept {
        return (reinterpret_cast<std::uintptr_t>(p) % Align) == 0;
    }

    template<class Target>
    inline Target& vector_cast(auto &&p) {
        return *reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p));
    }
}
#pragma GCC pop_options

#line 7 "cp-algo/util/complex.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo {
    // Custom implementation, since std::complex is UB on non-floating types
    template<typename T>
    struct complex {
        using value_type = T;
        T x, y;
        inline constexpr complex(): x(), y() {}
        inline constexpr complex(T const& x): x(x), y() {}
        inline constexpr complex(T const& x, T const& y): x(x), y(y) {}
        inline complex& operator *= (T const& t) {x *= t; y *= t; return *this;}
        inline complex& operator /= (T const& t) {x /= t; y /= t; return *this;}
        inline complex operator * (T const& t) const {return complex(*this) *= t;}
        inline complex operator / (T const& t) const {return complex(*this) /= t;}
        inline complex& operator += (complex const& t) {x += t.x; y += t.y; return *this;}
        inline complex& operator -= (complex const& t) {x -= t.x; y -= t.y; return *this;}
        inline complex operator * (complex const& t) const {return {x * t.x - y * t.y, x * t.y + y * t.x};}
        inline complex operator / (complex const& t) const {return *this * t.conj() / t.norm();}
        inline complex operator + (complex const& t) const {return complex(*this) += t;}
        inline complex operator - (complex const& t) const {return complex(*this) -= t;}
        inline complex& operator *= (complex const& t) {return *this = *this * t;}
        inline complex& operator /= (complex const& t) {return *this = *this / t;}
        inline complex operator - () const {return {-x, -y};}
        inline complex conj() const {return {x, -y};}
        inline T norm() const {return x * x + y * y;}
        inline T abs() const {return std::sqrt(norm());}
        inline T const real() const {return x;}
        inline T const imag() const {return y;}
        inline T& real() {return x;}
        inline T& imag() {return y;}
        inline static constexpr complex polar(T r, T theta) {return {T(r * cos(theta)), T(r * sin(theta))};}
        inline auto operator <=> (complex const& t) const = default;
    };
    template<typename T> inline complex<T> conj(complex<T> const& x) {return x.conj();}
    template<typename T> inline T norm(complex<T> const& x) {return x.norm();}
    template<typename T> inline T abs(complex<T> const& x) {return x.abs();}
    template<typename T> inline T& real(complex<T> &x) {return x.real();}
    template<typename T> inline T& imag(complex<T> &x) {return x.imag();}
    template<typename T> inline T const real(complex<T> const& x) {return x.real();}
    template<typename T> inline T const imag(complex<T> const& x) {return x.imag();}
    template<typename T>
    inline constexpr complex<T> polar(T r, T theta) {
        return complex<T>::polar(r, theta);
    }
    template<typename T>
    inline std::ostream& operator << (std::ostream &out, complex<T> const& x) {
        return out << x.real() << ' ' << x.imag();
    }
}
#pragma GCC pop_options

#line 7 "cp-algo/number_theory/two_squares.hpp"
#include <utility>
#line 10 "cp-algo/number_theory/two_squares.hpp"
namespace cp_algo::math {
    template<typename T>
    using gaussint = complex<T>;
    template<typename _Int>
    auto two_squares_prime_any(_Int p) {
        if(p == 2) {
            return gaussint<_Int>{1, 1};
        }
        assert(p % 4 == 1);
        using Int = std::make_signed_t<_Int>;
        using base = dynamic_modint<Int>;
        return base::with_mod(p, [&](){
            base g = primitive_root(p);
            int64_t i = bpow(g, (p - 1) / 4).getr();
            int64_t q0 = 1, q1 = 0;
            int64_t r = i, m = p;
            // TODO: Use library contfrac?
            do {
                int64_t d = r / m;
                q0 = std::exchange(q1, q0 + d * q1);
                r = std::exchange(m, r % m);
            } while(q1 < p / q1);
            return gaussint<_Int>{q0, (base(i) * base(q0)).rem()};
        });
    }

    template<typename Int>
    big_vector<gaussint<Int>> two_squares_all(Int n) {
        if(n == 0) {
            return {0};
        }
        auto primes = factorize(n);
        big_map<Int, int> cnt;
        for(auto p: primes) {
            cnt[p]++;
        }
        big_vector<gaussint<Int>> res = {1};
        for(auto [p, c]: cnt) {
            big_vector<gaussint<Int>> nres;
            if(p % 4 == 3) {
                if(c % 2 == 0) {
                    auto mul = bpow(gaussint<Int>(p), c / 2);
                    for(auto p: res) {
                        nres.push_back(p * mul);
                    }
                }
            } else if(p % 4 == 1) {
                auto base = two_squares_prime_any(p);
                for(int i = 0; i <= c; i++) {
                    auto mul = bpow(base, i) * bpow(conj(base), c - i);
                    for(auto p: res) {
                        nres.push_back(p * mul);
                    }
                }
            } else if(p % 4 == 2) {
                auto mul = bpow(gaussint<Int>(1, 1), c);
                for(auto p: res) {
                    nres.push_back(p * mul);
                }
            }
            res = nres;
        }
        big_vector<gaussint<Int>> nres;
        for(auto p: res) {
            while(p.real() < 0 || p.imag() < 0) {
                p *= gaussint<Int>(0, 1);
            }
            nres.push_back(p);
            if(!p.real() || !p.imag()) {
                nres.emplace_back(p.imag(), p.real());
            }
        }
        return nres;
    }
}

#ifndef CP_ALGO_NUMBER_THEORY_TWO_SQUARES_HPP
#define CP_ALGO_NUMBER_THEORY_TWO_SQUARES_HPP
#include "euler.hpp"
#include "../util/complex.hpp"
#include "../util/big_alloc.hpp"
#include <cassert>
#include <utility>
#include <vector>
#include <map>
namespace cp_algo::math{template<typename T>using gaussint=complex<T>;template<typename _Int>auto two_squares_prime_any(_Int p){if(p==2){return gaussint<_Int>{1,1};}assert(p%4==1);using Int=std::make_signed_t<_Int>;using base=dynamic_modint<Int>;return base::with_mod(p,[&](){base g=primitive_root(p);int64_t i=bpow(g,(p-1)/4).getr();int64_t q0=1,q1=0;int64_t r=i,m=p;do{int64_t d=r/m;q0=std::exchange(q1,q0+d*q1);r=std::exchange(m,r%m);}while(q1<p/q1);return gaussint<_Int>{q0,(base(i)*base(q0)).rem()};});}template<typename Int>big_vector<gaussint<Int>>two_squares_all(Int n){if(n==0){return{0};}auto primes=factorize(n);big_map<Int,int>cnt;for(auto p:primes){cnt[p]++;}big_vector<gaussint<Int>>res={1};for(auto[p,c]:cnt){big_vector<gaussint<Int>>nres;if(p%4==3){if(c%2==0){auto mul=bpow(gaussint<Int>(p),c/2);for(auto p:res){nres.push_back(p*mul);}}}else if(p%4==1){auto base=two_squares_prime_any(p);for(int i=0;i<=c;i++){auto mul=bpow(base,i)*bpow(conj(base),c-i);for(auto p:res){nres.push_back(p*mul);}}}else if(p%4==2){auto mul=bpow(gaussint<Int>(1,1),c);for(auto p:res){nres.push_back(p*mul);}}res=nres;}big_vector<gaussint<Int>>nres;for(auto p:res){while(p.real()<0||p.imag()<0){p*=gaussint<Int>(0,1);}nres.push_back(p);if(!p.real()||!p.imag()){nres.emplace_back(p.imag(),p.real());}}return nres;}}
#endif

#line 1 "cp-algo/number_theory/two_squares.hpp"
#line 1 "cp-algo/number_theory/euler.hpp"
#line 1 "cp-algo/number_theory/factorize.hpp"
#line 1 "cp-algo/number_theory/primality.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
#include <cassert>
namespace cp_algo::math{
#ifdef CP_ALGO_MAXN
const int maxn=CP_ALGO_MAXN;
#else
const int maxn=1<<19;
#endif
const int magic=64;auto bpow(auto const&x,auto n,auto const&one,auto op){if(n==0){return one;}else{auto t=bpow(x,n/2,one,op);t=op(t,t);if(n%2){t=op(t,x);}return t;}}auto bpow(auto x,auto n,auto ans){return bpow(x,n,ans,std::multiplies{});}template<typename T>T bpow(T const&x,auto n){return bpow(x,n,T(1));}inline constexpr auto inv2(auto x){assert(x%2);std::make_unsigned_t<decltype(x)>y=1;while(y*x!=1){y*=2-x*y;}return y;}}
#line 4 "cp-algo/number_theory/modint.hpp"
#include <iostream>
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math{template<typename modint,typename _Int>struct modint_base{using Int=_Int;using UInt=std::make_unsigned_t<Int>;static constexpr size_t bits=sizeof(Int)*8;using Int2=std::conditional_t<bits<=32,int64_t,__int128_t>;using UInt2=std::conditional_t<bits<=32,uint64_t,__uint128_t>;constexpr static Int mod(){return modint::mod();}constexpr static Int remod(){return modint::remod();}constexpr static UInt2 modmod(){return UInt2(mod())*mod();}constexpr modint_base()=default;constexpr modint_base(Int2 rr){to_modint().setr(UInt((rr+modmod())%mod()));}modint inv()const{return bpow(to_modint(),mod()-2);}modint operator-()const{modint neg;neg.r=std::min(-r,remod()-r);return neg;}modint&operator/=(const modint&t){return to_modint()*=t.inv();}modint&operator*=(const modint&t){r=UInt(UInt2(r)*t.r%mod());return to_modint();}modint&operator+=(const modint&t){r+=t.r;r=std::min(r,r-remod());return to_modint();}modint&operator-=(const modint&t){r-=t.r;r=std::min(r,r+remod());return to_modint();}modint operator+(const modint&t)const{return modint(to_modint())+=t;}modint operator-(const modint&t)const{return modint(to_modint())-=t;}modint operator*(const modint&t)const{return modint(to_modint())*=t;}modint operator/(const modint&t)const{return modint(to_modint())/=t;}auto operator==(const modint&t)const{return to_modint().getr()==t.getr();}auto operator!=(const modint&t)const{return to_modint().getr()!=t.getr();}auto operator<=(const modint&t)const{return to_modint().getr()<=t.getr();}auto operator>=(const modint&t)const{return to_modint().getr()>=t.getr();}auto operator<(const modint&t)const{return to_modint().getr()<t.getr();}auto operator>(const modint&t)const{return to_modint().getr()>t.getr();}Int rem()const{UInt R=to_modint().getr();return R-(R>(UInt)mod()/2)*mod();}constexpr void setr(UInt rr){r=rr;}constexpr UInt getr()const{return r;}static UInt modmod8(){return UInt(8*modmod());}void add_unsafe(UInt t){r+=t;}void pseudonormalize(){r=std::min(r,r-modmod8());}modint const&normalize(){if(r>=(UInt)mod()){r%=mod();}return to_modint();}void setr_direct(UInt rr){r=rr;}UInt getr_direct()const{return r;}protected:UInt r;private:constexpr modint&to_modint(){return static_cast<modint&>(*this);}constexpr modint const&to_modint()const{return static_cast<modint const&>(*this);}};template<typename modint>concept modint_type=std::is_base_of_v<modint_base<modint,typename modint::Int>,modint>;template<modint_type modint>decltype(std::cin)&operator>>(decltype(std::cin)&in,modint&x){typename modint::UInt r;auto&res=in>>r;x.setr(r);return res;}template<modint_type modint>decltype(std::cout)&operator<<(decltype(std::cout)&out,modint const&x){return out<<x.getr();}template<auto m>struct modint:modint_base<modint<m>,decltype(m)>{using Base=modint_base<modint<m>,decltype(m)>;using Base::Base;static constexpr Base::Int mod(){return m;}static constexpr Base::UInt remod(){return m;}auto getr()const{return Base::r;}};template<typename Int=int>struct dynamic_modint:modint_base<dynamic_modint<Int>,Int>{using Base=modint_base<dynamic_modint<Int>,Int>;using Base::Base;static Base::UInt m_reduce(Base::UInt2 ab){if(mod()%2==0)[[unlikely]]{return typename Base::UInt(ab%mod());}else{typename Base::UInt2 m=typename Base::UInt(ab)*imod();return typename Base::UInt((ab+m*mod())>>Base::bits);}}static Base::UInt m_transform(Base::UInt a){if(mod()%2==0)[[unlikely]]{return a;}else{return m_reduce(a*pw128());}}dynamic_modint&operator*=(const dynamic_modint&t){Base::r=m_reduce(typename Base::UInt2(Base::r)*t.r);return*this;}void setr(Base::UInt rr){Base::r=m_transform(rr);}Base::UInt getr()const{typename Base::UInt res=m_reduce(Base::r);return std::min(res,res-mod());}static Int mod(){return m;}static Int remod(){return 2*m;}static Base::UInt imod(){return im;}static Base::UInt2 pw128(){return r2;}static void switch_mod(Int nm){m=nm;im=m%2?inv2(-m):0;r2=static_cast<Base::UInt>(static_cast<Base::UInt2>(-1)%m+1);}auto static with_mod(Int tmp,auto callback){struct scoped{Int prev=mod();~scoped(){switch_mod(prev);}}_;switch_mod(tmp);return callback();}private:static thread_local Int m;static thread_local Base::UInt im,r2;};template<typename Int>Int thread_local dynamic_modint<Int>::m=1;template<typename Int>dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::im=-1;template<typename Int>dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::r2=0;}
#line 4 "cp-algo/number_theory/primality.hpp"
#include <algorithm>
#include <bit>
namespace cp_algo::math{template<typename _Int>bool is_prime(_Int m){using Int=std::make_signed_t<_Int>;using UInt=std::make_unsigned_t<Int>;if(m==1||m%2==0){return m==2;}int s=std::countr_zero(UInt(m-1));auto d=(m-1)>>s;using base=dynamic_modint<Int>;auto test=[&](base x){x=bpow(x,d);if(std::abs(x.rem())<=1){return true;}for(int i=1;i<s&&x!=-1;i++){x*=x;}return x==-1;};return base::with_mod(m,[&](){
#ifdef CP_ALGO_NUMBER_THEORY_PRIMALITY_BASES_HPP
uint16_t base2=7,base3=61;if(m!=uint32_t(m)){base2=base_table1[uint32_t(m*0xAD625B89)>>18];base3=base_table2[base2>>13];}return test(2)&&test(base2)&&test(base3);
#else
return std::ranges::all_of(std::array{2,325,9375,28178,450775,9780504,1795265022},test);
#endif
});}}
#line 1 "cp-algo/util/big_alloc.hpp"
#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#include <vector>
#include <string>
#include <cstddef>
#line 13 "cp-algo/util/big_alloc.hpp"
#include <generator>
#include <forward_list>
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
#  define CP_ALGO_USE_MMAP 1
#  include <sys/mman.h>
#else
#  define CP_ALGO_USE_MMAP 0
#endif
namespace cp_algo{template<typename T,size_t Align=32>class big_alloc{static_assert(Align>=alignof(void*),"Align must be at least pointer-size");static_assert(std::popcount(Align)==1,"Align must be a power of two");public:using value_type=T;template<class U>struct rebind{using other=big_alloc<U,Align>;};constexpr bool operator==(const big_alloc&)const=default;constexpr bool operator!=(const big_alloc&)const=default;big_alloc()noexcept=default;template<typename U,std::size_t A>big_alloc(const big_alloc<U,A>&)noexcept{}[[nodiscard]]T*allocate(std::size_t n){std::size_t padded=round_up(n*sizeof(T));std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){void*raw=mmap(nullptr,padded,PROT_READ|PROT_WRITE,MAP_PRIVATE|MAP_ANONYMOUS,-1,0);madvise(raw,padded,MADV_HUGEPAGE);madvise(raw,padded,MADV_POPULATE_WRITE);return static_cast<T*>(raw);}
#endif
return static_cast<T*>(::operator new(padded,std::align_val_t(align)));}void deallocate(T*p,std::size_t n)noexcept{if(!p)return;std::size_t padded=round_up(n*sizeof(T));std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){munmap(p,padded);return;}
#endif
::operator delete(p,padded,std::align_val_t(align));}private:static constexpr std::size_t MEGABYTE=1<<20;static constexpr std::size_t round_up(std::size_t x)noexcept{return(x+Align-1)/Align*Align;}};template<typename T>using big_vector=std::vector<T,big_alloc<T>>;template<typename T>using big_basic_string=std::basic_string<T,std::char_traits<T>,big_alloc<T>>;template<typename T>using big_deque=std::deque<T,big_alloc<T>>;template<typename T>using big_stack=std::stack<T,big_deque<T>>;template<typename T>using big_queue=std::queue<T,big_deque<T>>;template<typename T>using big_priority_queue=std::priority_queue<T,big_vector<T>>;template<typename T>using big_forward_list=std::forward_list<T,big_alloc<T>>;using big_string=big_basic_string<char>;template<typename Key,typename Value,typename Compare=std::less<Key>>using big_map=std::map<Key,Value,Compare,big_alloc<std::pair<const Key,Value>>>;template<typename T,typename Compare=std::less<T>>using big_multiset=std::multiset<T,Compare,big_alloc<T>>;template<typename T,typename Compare=std::less<T>>using big_set=std::set<T,Compare,big_alloc<T>>;template<typename Ref,typename V=void>using big_generator=std::generator<Ref,V,big_alloc<std::byte>>;}namespace std::ranges{template<typename Ref,typename V>elements_of(cp_algo::big_generator<Ref,V>&&)->elements_of<cp_algo::big_generator<Ref,V>&&,cp_algo::big_alloc<std::byte>>;}
#line 1 "cp-algo/random/rng.hpp"
#include <chrono>
#include <random>
namespace cp_algo::random{std::mt19937_64 gen(std::chrono::steady_clock::now().time_since_epoch().count());uint64_t rng(){return gen();}}
#line 7 "cp-algo/number_theory/factorize.hpp"
namespace cp_algo::math{template<typename _Int>auto proper_divisor(_Int m){using Int=std::make_signed_t<_Int>;using base=dynamic_modint<Int>;return m%2==0?2:base::with_mod(m,[&](){base t=random::rng();auto f=[&](auto x){return x*x+t;};base x=0,y=0;base g=1;while(g==1){for(int i=0;i<64;i++){x=f(x);y=f(f(y));if(x==y)[[unlikely]]{t=random::rng();x=y=0;}else{base t=g*(x-y);g=t==0?g:t;}}g=std::gcd(g.getr(),m);}return g.getr();});}template<typename Int>big_generator<Int>factorize(Int m){if(is_prime(m)){co_yield m;}else if(m>1){auto g=proper_divisor(m);co_yield std::ranges::elements_of(factorize(g));co_yield std::ranges::elements_of(factorize(m/g));}}template<typename Int>big_generator<Int>divisors_sqrt(Int m){for(Int i=1;i*i<=m;i++){if(m%i==0){co_yield i;if(i*i!=m){co_yield m/i;}}}}}
#line 5 "cp-algo/number_theory/euler.hpp"
namespace cp_algo::math{auto euler_phi(auto m){using T=std::decay_t<decltype(m)>;auto primes=to<big_vector<T>>(factorize(m));std::ranges::sort(primes);auto[from,to]=std::ranges::unique(primes);primes.erase(from,to);auto ans=m;for(auto it:primes){ans-=ans/it;}return ans;}template<modint_type base>auto period(base x){auto ans=euler_phi(base::mod());base x0=bpow(x,ans);for(auto t:factorize(ans)){while(ans%t==0&&x0*bpow(x,ans/t)==x0){ans/=t;}}return ans;}template<typename _Int>_Int primitive_root(_Int p){using Int=std::make_signed_t<_Int>;using base=dynamic_modint<Int>;return base::with_mod(p,[p](){base t=1;while(period(t)!=p-1){t=random::rng();}return t.getr();});}}
#line 1 "cp-algo/util/complex.hpp"
#line 4 "cp-algo/util/complex.hpp"
#include <cmath>
#include <type_traits>
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#line 6 "cp-algo/util/simd.hpp"
#include <memory>
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_SIMD_AVX2_TARGET _Pragma("GCC target(\"avx2\")")
#else
#define CP_ALGO_SIMD_AVX2_TARGET
#endif
#define CP_ALGO_SIMD_PRAGMA_PUSH \
_Pragma("GCC push_options")\CP_ALGO_SIMD_AVX2_TARGETCP_ALGO_SIMD_PRAGMA_PUSHnamespace cp_algo{template<typename T,size_t len>using simd[[gnu::vector_size(len*sizeof(T))]]=T;using u64x8=simd<uint64_t,8>;using u32x16=simd<uint32_t,16>;using i64x4=simd<int64_t,4>;using u64x4=simd<uint64_t,4>;using u32x8=simd<uint32_t,8>;using u16x16=simd<uint16_t,16>;using i32x4=simd<int32_t,4>;using u32x4=simd<uint32_t,4>;using u16x8=simd<uint16_t,8>;using u16x4=simd<uint16_t,4>;using i16x4=simd<int16_t,4>;using u8x32=simd<uint8_t,32>;using u8x8=simd<uint8_t,8>;using u8x4=simd<uint8_t,4>;using dx4=simd<double,4>;inline dx4 abs(dx4 a){return dx4{std::abs(a[0]),std::abs(a[1]),std::abs(a[2]),std::abs(a[3])};}static constexpr dx4 magic=dx4()+(3ULL<<51);inline i64x4 lround(dx4 x){return i64x4(x+magic)-i64x4(magic);}inline dx4 to_double(i64x4 x){return dx4(x+i64x4(magic))-magic;}inline dx4 round(dx4 a){return dx4{std::nearbyint(a[0]),std::nearbyint(a[1]),std::nearbyint(a[2]),std::nearbyint(a[3])};}inline u64x4 low32(u64x4 x){return x&uint32_t(-1);}inline auto swap_bytes(auto x){return decltype(x)(__builtin_shufflevector(u32x8(x),u32x8(x),1,0,3,2,5,4,7,6));}inline u64x4 montgomery_reduce(u64x4 x,uint32_t mod,uint32_t imod){
#ifdef __AVX2__
auto x_ninv=u64x4(_mm256_mul_epu32(__m256i(x),__m256i()+imod));x+=u64x4(_mm256_mul_epu32(__m256i(x_ninv),__m256i()+mod));
#else
auto x_ninv=u64x4(u32x8(low32(x))*imod);x+=x_ninv*uint64_t(mod);
#endif
return swap_bytes(x);}inline u64x4 montgomery_mul(u64x4 x,u64x4 y,uint32_t mod,uint32_t imod){
#ifdef __AVX2__
return montgomery_reduce(u64x4(_mm256_mul_epu32(__m256i(x),__m256i(y))),mod,imod);
#else
return montgomery_reduce(x*y,mod,imod);
#endif
}inline u32x8 montgomery_mul(u32x8 x,u32x8 y,uint32_t mod,uint32_t imod){return u32x8(montgomery_mul(u64x4(x),u64x4(y),mod,imod))|u32x8(swap_bytes(montgomery_mul(u64x4(swap_bytes(x)),u64x4(swap_bytes(y)),mod,imod)));}inline dx4 rotate_right(dx4 x){static constexpr u64x4 shuffler={3,0,1,2};return __builtin_shuffle(x,shuffler);}template<std::size_t Align=32>inline bool is_aligned(const auto*p)noexcept{return(reinterpret_cast<std::uintptr_t>(p)%Align)==0;}template<class Target>inline Target&vector_cast(auto&&p){return*reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p));}}
#pragma GCC pop_options
#line 7 "cp-algo/util/complex.hpp"
CP_ALGO_SIMD_PRAGMA_PUSHnamespace cp_algo{template<typename T>struct complex{using value_type=T;T x,y;inline constexpr complex():x(),y(){}inline constexpr complex(T const&x):x(x),y(){}inline constexpr complex(T const&x,T const&y):x(x),y(y){}inline complex&operator*=(T const&t){x*=t;y*=t;return*this;}inline complex&operator/=(T const&t){x/=t;y/=t;return*this;}inline complex operator*(T const&t)const{return complex(*this)*=t;}inline complex operator/(T const&t)const{return complex(*this)/=t;}inline complex&operator+=(complex const&t){x+=t.x;y+=t.y;return*this;}inline complex&operator-=(complex const&t){x-=t.x;y-=t.y;return*this;}inline complex operator*(complex const&t)const{return{x*t.x-y*t.y,x*t.y+y*t.x};}inline complex operator/(complex const&t)const{return*this*t.conj()/t.norm();}inline complex operator+(complex const&t)const{return complex(*this)+=t;}inline complex operator-(complex const&t)const{return complex(*this)-=t;}inline complex&operator*=(complex const&t){return*this=*this*t;}inline complex&operator/=(complex const&t){return*this=*this/t;}inline complex operator-()const{return{-x,-y};}inline complex conj()const{return{x,-y};}inline T norm()const{return x*x+y*y;}inline T abs()const{return std::sqrt(norm());}inline T const real()const{return x;}inline T const imag()const{return y;}inline T&real(){return x;}inline T&imag(){return y;}inline static constexpr complex polar(T r,T theta){return{T(r*cos(theta)),T(r*sin(theta))};}inline auto operator<=>(complex const&t)const=default;};template<typename T>inline complex<T>conj(complex<T>const&x){return x.conj();}template<typename T>inline T norm(complex<T>const&x){return x.norm();}template<typename T>inline T abs(complex<T>const&x){return x.abs();}template<typename T>inline T&real(complex<T>&x){return x.real();}template<typename T>inline T&imag(complex<T>&x){return x.imag();}template<typename T>inline T const real(complex<T>const&x){return x.real();}template<typename T>inline T const imag(complex<T>const&x){return x.imag();}template<typename T>inline constexpr complex<T>polar(T r,T theta){return complex<T>::polar(r,theta);}template<typename T>inline std::ostream&operator<<(std::ostream&out,complex<T>const&x){return out<<x.real()<<' '<<x.imag();}}
#pragma GCC pop_options
#line 7 "cp-algo/number_theory/two_squares.hpp"
#include <utility>
#line 10 "cp-algo/number_theory/two_squares.hpp"
namespace cp_algo::math{template<typename T>using gaussint=complex<T>;template<typename _Int>auto two_squares_prime_any(_Int p){if(p==2){return gaussint<_Int>{1,1};}assert(p%4==1);using Int=std::make_signed_t<_Int>;using base=dynamic_modint<Int>;return base::with_mod(p,[&](){base g=primitive_root(p);int64_t i=bpow(g,(p-1)/4).getr();int64_t q0=1,q1=0;int64_t r=i,m=p;do{int64_t d=r/m;q0=std::exchange(q1,q0+d*q1);r=std::exchange(m,r%m);}while(q1<p/q1);return gaussint<_Int>{q0,(base(i)*base(q0)).rem()};});}template<typename Int>big_vector<gaussint<Int>>two_squares_all(Int n){if(n==0){return{0};}auto primes=factorize(n);big_map<Int,int>cnt;for(auto p:primes){cnt[p]++;}big_vector<gaussint<Int>>res={1};for(auto[p,c]:cnt){big_vector<gaussint<Int>>nres;if(p%4==3){if(c%2==0){auto mul=bpow(gaussint<Int>(p),c/2);for(auto p:res){nres.push_back(p*mul);}}}else if(p%4==1){auto base=two_squares_prime_any(p);for(int i=0;i<=c;i++){auto mul=bpow(base,i)*bpow(conj(base),c-i);for(auto p:res){nres.push_back(p*mul);}}}else if(p%4==2){auto mul=bpow(gaussint<Int>(1,1),c);for(auto p:res){nres.push_back(p*mul);}}res=nres;}big_vector<gaussint<Int>>nres;for(auto p:res){while(p.real()<0||p.imag()<0){p*=gaussint<Int>(0,1);}nres.push_back(p);if(!p.real()||!p.imag()){nres.emplace_back(p.imag(),p.real());}}return nres;}}

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