#include <algorithm>
#include <array>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

// @param n `0 <= n`
// @return minimum non-negative `x` s.t. `n <= 2**x`
int ceil_pow2(int n) {
    int x = 0;
    while ((1U << x) < (unsigned int)(n)) x++;
    return x;
}

// @param n `1 <= n`
// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
int bsf(unsigned int n) {
#ifdef _MSC_VER
    unsigned long index;
    _BitScanForward(&index, n);
    return index;
#else
    return __builtin_ctz(n);
#endif
}

}  // namespace internal

}  // namespace atcoder



#include <utility>

namespace atcoder {

namespace internal {

// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
    unsigned int _m;
    unsigned long long im;

    // @param m `1 <= m < 2^31`
    barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

    // @return m
    unsigned int umod() const { return _m; }

    // @param a `0 <= a < m`
    // @param b `0 <= b < m`
    // @return `a * b % m`
    unsigned int mul(unsigned int a, unsigned int b) const {
        // [1] m = 1
        // a = b = im = 0, so okay

        // [2] m >= 2
        // im = ceil(2^64 / m)
        // -> im * m = 2^64 + r (0 <= r < m)
        // let z = a*b = c*m + d (0 <= c, d < m)
        // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
        // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
        // ((ab * im) >> 64) == c or c + 1
        unsigned long long z = a;
        z *= b;
#ifdef _MSC_VER
        unsigned long long x;
        _umul128(z, im, &x);
#else
        unsigned long long x =
            (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
        unsigned int v = (unsigned int)(z - x * _m);
        if (_m <= v) v += _m;
        return v;
    }
};

// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
    if (m == 1) return 0;
    unsigned int _m = (unsigned int)(m);
    unsigned long long r = 1;
    unsigned long long y = safe_mod(x, m);
    while (n) {
        if (n & 1) r = (r * y) % _m;
        y = (y * y) % _m;
        n >>= 1;
    }
    return r;
}

// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
    if (n <= 1) return false;
    if (n == 2 || n == 7 || n == 61) return true;
    if (n % 2 == 0) return false;
    long long d = n - 1;
    while (d % 2 == 0) d /= 2;
    constexpr long long bases[3] = {2, 7, 61};
    for (long long a : bases) {
        long long t = d;
        long long y = pow_mod_constexpr(a, t, n);
        while (t != n - 1 && y != 1 && y != n - 1) {
            y = y * y % n;
            t <<= 1;
        }
        if (y != n - 1 && t % 2 == 0) {
            return false;
        }
    }
    return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);

// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
    a = safe_mod(a, b);
    if (a == 0) return {b, 0};

    // Contracts:
    // [1] s - m0 * a = 0 (mod b)
    // [2] t - m1 * a = 0 (mod b)
    // [3] s * |m1| + t * |m0| <= b
    long long s = b, t = a;
    long long m0 = 0, m1 = 1;

    while (t) {
        long long u = s / t;
        s -= t * u;
        m0 -= m1 * u;  // |m1 * u| <= |m1| * s <= b

        // [3]:
        // (s - t * u) * |m1| + t * |m0 - m1 * u|
        // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
        // = s * |m1| + t * |m0| <= b

        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    // by [3]: |m0| <= b/g
    // by g != b: |m0| < b/g
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}

// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
    if (m == 2) return 1;
    if (m == 167772161) return 3;
    if (m == 469762049) return 3;
    if (m == 754974721) return 11;
    if (m == 998244353) return 3;
    int divs[20] = {};
    divs[0] = 2;
    int cnt = 1;
    int x = (m - 1) / 2;
    while (x % 2 == 0) x /= 2;
    for (int i = 3; (long long)(i)*i <= x; i += 2) {
        if (x % i == 0) {
            divs[cnt++] = i;
            while (x % i == 0) {
                x /= i;
            }
        }
    }
    if (x > 1) {
        divs[cnt++] = x;
    }
    for (int g = 2;; g++) {
        bool ok = true;
        for (int i = 0; i < cnt; i++) {
            if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
                ok = false;
                break;
            }
        }
        if (ok) return g;
    }
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

}  // namespace internal

}  // namespace atcoder


#include <cassert>
#include <numeric>
#include <type_traits>

namespace atcoder {

namespace internal {

#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value ||
                                  std::is_same<T, __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int128 =
    typename std::conditional<std::is_same<T, __uint128_t>::value ||
                                  std::is_same<T, unsigned __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using make_unsigned_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value,
                              __uint128_t,
                              unsigned __int128>;

template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
                                                  is_signed_int128<T>::value ||
                                                  is_unsigned_int128<T>::value,
                                              std::true_type,
                                              std::false_type>::type;

template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
                                                 std::is_signed<T>::value) ||
                                                    is_signed_int128<T>::value,
                                                std::true_type,
                                                std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_unsigned<T>::value) ||
                                  is_unsigned_int128<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<
    is_signed_int128<T>::value,
    make_unsigned_int128<T>,
    typename std::conditional<std::is_signed<T>::value,
                              std::make_unsigned<T>,
                              std::common_type<T>>::type>::type;

#else

template <class T> using is_integral = typename std::is_integral<T>;

template <class T>
using is_signed_int =
    typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<is_integral<T>::value &&
                                  std::is_unsigned<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
                                              std::make_unsigned<T>,
                                              std::common_type<T>>::type;

#endif

template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

template <class T> using to_unsigned_t = typename to_unsigned<T>::type;

}  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <numeric>
#include <sstream>
#include <string>
#include <type_traits>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

	struct modint_base {};
	struct static_modint_base: modint_base {};

	template<class T>
	using is_modint = std::is_base_of<modint_base, T>;
	template<class T>
	using is_modint_t
		= std::enable_if_t<is_modint<T>::value>;

}  // namespace internal

template<int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint: internal::static_modint_base {
	using mint = static_modint;

public:
	static constexpr int mod() {
		return m;
	}
	static mint raw(int v) {
		mint x;
		x._v = v;
		return x;
	}

	static_modint(): _v(0) {
	}
	template<class T,
			 internal::is_signed_int_t<T>* = nullptr>
	static_modint(T v) {
		long long x
			= (long long) (v % (long long) (umod()));
		if(x < 0) x += umod();
		_v = (unsigned int) (x);
	}
	template<class T,
			 internal::is_unsigned_int_t<T>* = nullptr>
	static_modint(T v) {
		_v = (unsigned int) (v % umod());
	}
	static_modint(bool v) {
		_v = ((unsigned int) (v) % umod());
	}

	unsigned int val() const {
		return _v;
	}

	mint& operator++() {
		_v++;
		if(_v == umod()) _v = 0;
		return *this;
	}
	mint& operator--() {
		if(_v == 0) _v = umod();
		_v--;
		return *this;
	}
	mint operator++(int) {
		mint result = *this;
		++*this;
		return result;
	}
	mint operator--(int) {
		mint result = *this;
		--*this;
		return result;
	}

	mint& operator+=(const mint& rhs) {
		_v += rhs._v;
		if(_v >= umod()) _v -= umod();
		return *this;
	}
	mint& operator-=(const mint& rhs) {
		_v -= rhs._v;
		if(_v >= umod()) _v += umod();
		return *this;
	}
	mint& operator*=(const mint& rhs) {
		unsigned long long z = _v;
		z *= rhs._v;
		_v = (unsigned int) (z % umod());
		return *this;
	}
	mint& operator/=(const mint& rhs) {
		return *this = *this * rhs.inv();
	}

	mint operator+() const {
		return *this;
	}
	mint operator-() const {
		return mint() - *this;
	}

	mint pow(long long n) const {
		assert(0 <= n);
		mint x = *this, r = 1;
		while(n) {
			if(n & 1) r *= x;
			x *= x;
			n >>= 1;
		}
		return r;
	}
	mint inv() const {
		if(prime) {
			assert(_v);
			return pow(umod() - 2);
		} else {
			auto eg = internal::inv_gcd(_v, m);
			assert(eg.first == 1);
			return eg.second;
		}
	}

	friend mint operator+(const mint& lhs,
						  const mint& rhs) {
		return mint(lhs) += rhs;
	}
	friend mint operator-(const mint& lhs,
						  const mint& rhs) {
		return mint(lhs) -= rhs;
	}
	friend mint operator*(const mint& lhs,
						  const mint& rhs) {
		return mint(lhs) *= rhs;
	}
	friend mint operator/(const mint& lhs,
						  const mint& rhs) {
		return mint(lhs) /= rhs;
	}
	friend bool operator==(const mint& lhs,
						   const mint& rhs) {
		return lhs._v == rhs._v;
	}
	friend bool operator!=(const mint& lhs,
						   const mint& rhs) {
		return lhs._v != rhs._v;
	}
	std::string dump() const {
		std::stringstream ss;
		ss << val();
		return ss.str();
	}

private:
	unsigned int _v;
	static constexpr unsigned int umod() {
		return m;
	}
	static constexpr bool prime = internal::is_prime<m>;
};

template<int id>
struct dynamic_modint: internal::modint_base {
	using mint = dynamic_modint;

public:
	static int mod() {
		return (int) (bt.umod());
	}
	static void set_mod(int m) {
		assert(1 <= m);
		bt = internal::barrett(m);
	}
	static mint raw(int v) {
		mint x;
		x._v = v;
		return x;
	}

	dynamic_modint(): _v(0) {
	}
	template<class T,
			 internal::is_signed_int_t<T>* = nullptr>
	dynamic_modint(T v) {
		long long x = (long long) (v % (long long) (mod()));
		if(x < 0) x += mod();
		_v = (unsigned int) (x);
	}
	template<class T,
			 internal::is_unsigned_int_t<T>* = nullptr>
	dynamic_modint(T v) {
		_v = (unsigned int) (v % mod());
	}
	dynamic_modint(bool v) {
		_v = ((unsigned int) (v) % mod());
	}

	unsigned int val() const {
		return _v;
	}

	mint& operator++() {
		_v++;
		if(_v == umod()) _v = 0;
		return *this;
	}
	mint& operator--() {
		if(_v == 0) _v = umod();
		_v--;
		return *this;
	}
	mint operator++(int) {
		mint result = *this;
		++*this;
		return result;
	}
	mint operator--(int) {
		mint result = *this;
		--*this;
		return result;
	}

	mint& operator+=(const mint& rhs) {
		_v += rhs._v;
		if(_v >= umod()) _v -= umod();
		return *this;
	}
	mint& operator-=(const mint& rhs) {
		_v += mod() - rhs._v;
		if(_v >= umod()) _v -= umod();
		return *this;
	}
	mint& operator*=(const mint& rhs) {
		_v = bt.mul(_v, rhs._v);
		return *this;
	}
	mint& operator/=(const mint& rhs) {
		return *this = *this * rhs.inv();
	}

	mint operator+() const {
		return *this;
	}
	mint operator-() const {
		return mint() - *this;
	}

	mint pow(long long n) const {
		assert(0 <= n);
		mint x = *this, r = 1;
		while(n) {
			if(n & 1) r *= x;
			x *= x;
			n >>= 1;
		}
		return r;
	}
	mint inv() const {
		auto eg = internal::inv_gcd(_v, mod());
		assert(eg.first == 1);
		return eg.second;
	}

	friend mint operator+(const mint& lhs,
						  const mint& rhs) {
		return mint(lhs) += rhs;
	}
	friend mint operator-(const mint& lhs,
						  const mint& rhs) {
		return mint(lhs) -= rhs;
	}
	friend mint operator*(const mint& lhs,
						  const mint& rhs) {
		return mint(lhs) *= rhs;
	}
	friend mint operator/(const mint& lhs,
						  const mint& rhs) {
		return mint(lhs) /= rhs;
	}
	friend bool operator==(const mint& lhs,
						   const mint& rhs) {
		return lhs._v == rhs._v;
	}
	friend bool operator!=(const mint& lhs,
						   const mint& rhs) {
		return lhs._v != rhs._v;
	}
	std::string dump() const {
		std::stringstream ss;
		ss << "modint<" << mod() << ">(" << val() << ")";
		return ss.str();
	}

private:
	unsigned int _v;
	static internal::barrett bt;
	static unsigned int umod() {
		return bt.umod();
	}
};
template<int id>
internal::barrett dynamic_modint<id>::bt = 998244353;

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

namespace internal {

	template<class T>
	using is_static_modint
		= std::is_base_of<internal::static_modint_base, T>;

	template<class T>
	using is_static_modint_t
		= std::enable_if_t<is_static_modint<T>::value>;

	template<class>
	struct is_dynamic_modint: public std::false_type {};
	template<int id>
	struct is_dynamic_modint<dynamic_modint<id>>
		: public std::true_type {};

	template<class T>
	using is_dynamic_modint_t
		= std::enable_if_t<is_dynamic_modint<T>::value>;

}  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <type_traits>
#include <vector>

namespace atcoder {

namespace internal {

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly(std::vector<mint>& a) {
    static constexpr int g = internal::primitive_root<mint::mod()>;
    int n = int(a.size());
    int h = internal::ceil_pow2(n);

    static bool first = true;
    static mint sum_e[30];  // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
    if (first) {
        first = false;
        mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
        int cnt2 = bsf(mint::mod() - 1);
        mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
        for (int i = cnt2; i >= 2; i--) {
            // e^(2^i) == 1
            es[i - 2] = e;
            ies[i - 2] = ie;
            e *= e;
            ie *= ie;
        }
        mint now = 1;
        for (int i = 0; i <= cnt2 - 2; i++) {
            sum_e[i] = es[i] * now;
            now *= ies[i];
        }
    }
    for (int ph = 1; ph <= h; ph++) {
        int w = 1 << (ph - 1), p = 1 << (h - ph);
        mint now = 1;
        for (int s = 0; s < w; s++) {
            int offset = s << (h - ph + 1);
            for (int i = 0; i < p; i++) {
                auto l = a[i + offset];
                auto r = a[i + offset + p] * now;
                a[i + offset] = l + r;
                a[i + offset + p] = l - r;
            }
            now *= sum_e[bsf(~(unsigned int)(s))];
        }
    }
}

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly_inv(std::vector<mint>& a) {
    static constexpr int g = internal::primitive_root<mint::mod()>;
    int n = int(a.size());
    int h = internal::ceil_pow2(n);

    static bool first = true;
    static mint sum_ie[30];  // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
    if (first) {
        first = false;
        mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
        int cnt2 = bsf(mint::mod() - 1);
        mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
        for (int i = cnt2; i >= 2; i--) {
            // e^(2^i) == 1
            es[i - 2] = e;
            ies[i - 2] = ie;
            e *= e;
            ie *= ie;
        }
        mint now = 1;
        for (int i = 0; i <= cnt2 - 2; i++) {
            sum_ie[i] = ies[i] * now;
            now *= es[i];
        }
    }

    for (int ph = h; ph >= 1; ph--) {
        int w = 1 << (ph - 1), p = 1 << (h - ph);
        mint inow = 1;
        for (int s = 0; s < w; s++) {
            int offset = s << (h - ph + 1);
            for (int i = 0; i < p; i++) {
                auto l = a[i + offset];
                auto r = a[i + offset + p];
                a[i + offset] = l + r;
                a[i + offset + p] =
                    (unsigned long long)(mint::mod() + l.val() - r.val()) *
                    inow.val();
            }
            inow *= sum_ie[bsf(~(unsigned int)(s))];
        }
    }
}

}  // namespace internal

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(std::vector<mint> a, std::vector<mint> b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};
    if (std::min(n, m) <= 60) {
        if (n < m) {
            std::swap(n, m);
            std::swap(a, b);
        }
        std::vector<mint> ans(n + m - 1);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                ans[i + j] += a[i] * b[j];
            }
        }
        return ans;
    }
    int z = 1 << internal::ceil_pow2(n + m - 1);
    a.resize(z);
    internal::butterfly(a);
    b.resize(z);
    internal::butterfly(b);
    for (int i = 0; i < z; i++) {
        a[i] *= b[i];
    }
    internal::butterfly_inv(a);
    a.resize(n + m - 1);
    mint iz = mint(z).inv();
    for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
    return a;
}

template <unsigned int mod = 998244353,
          class T,
          std::enable_if_t<internal::is_integral<T>::value>* = nullptr>
std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};

    using mint = static_modint<mod>;
    std::vector<mint> a2(n), b2(m);
    for (int i = 0; i < n; i++) {
        a2[i] = mint(a[i]);
    }
    for (int i = 0; i < m; i++) {
        b2[i] = mint(b[i]);
    }
    auto c2 = convolution(move(a2), move(b2));
    std::vector<T> c(n + m - 1);
    for (int i = 0; i < n + m - 1; i++) {
        c[i] = c2[i].val();
    }
    return c;
}

std::vector<long long> convolution_ll(const std::vector<long long>& a,
                                      const std::vector<long long>& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};

    static constexpr unsigned long long MOD1 = 754974721;  // 2^24
    static constexpr unsigned long long MOD2 = 167772161;  // 2^25
    static constexpr unsigned long long MOD3 = 469762049;  // 2^26
    static constexpr unsigned long long M2M3 = MOD2 * MOD3;
    static constexpr unsigned long long M1M3 = MOD1 * MOD3;
    static constexpr unsigned long long M1M2 = MOD1 * MOD2;
    static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;

    static constexpr unsigned long long i1 =
        internal::inv_gcd(MOD2 * MOD3, MOD1).second;
    static constexpr unsigned long long i2 =
        internal::inv_gcd(MOD1 * MOD3, MOD2).second;
    static constexpr unsigned long long i3 =
        internal::inv_gcd(MOD1 * MOD2, MOD3).second;

    auto c1 = convolution<MOD1>(a, b);
    auto c2 = convolution<MOD2>(a, b);
    auto c3 = convolution<MOD3>(a, b);

    std::vector<long long> c(n + m - 1);
    for (int i = 0; i < n + m - 1; i++) {
        unsigned long long x = 0;
        x += (c1[i] * i1) % MOD1 * M2M3;
        x += (c2[i] * i2) % MOD2 * M1M3;
        x += (c3[i] * i3) % MOD3 * M1M2;
        // B = 2^63, -B <= x, r(real value) < B
        // (x, x - M, x - 2M, or x - 3M) = r (mod 2B)
        // r = c1[i] (mod MOD1)
        // focus on MOD1
        // r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B)
        // r = x,
        //     x - M' + (0 or 2B),
        //     x - 2M' + (0, 2B or 4B),
        //     x - 3M' + (0, 2B, 4B or 6B) (without mod!)
        // (r - x) = 0, (0)
        //           - M' + (0 or 2B), (1)
        //           -2M' + (0 or 2B or 4B), (2)
        //           -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1)
        // we checked that
        //   ((1) mod MOD1) mod 5 = 2
        //   ((2) mod MOD1) mod 5 = 3
        //   ((3) mod MOD1) mod 5 = 4
        long long diff =
            c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
        if (diff < 0) diff += MOD1;
        static constexpr unsigned long long offset[5] = {
            0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
        x -= offset[diff % 5];
        c[i] = x;
    }

    return c;
}

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <vector>

namespace atcoder {

// Implement (union by size) + (path compression)
// Reference:
// Zvi Galil and Giuseppe F. Italiano,
// Data structures and algorithms for disjoint set union problems
struct dsu {
public:
	dsu(): _n(0) {
	}
	dsu(int n): _n(n), parent_or_size(n, -1) {
	}

	int merge(int a, int b) {
		assert(0 <= a && a < _n);
		assert(0 <= b && b < _n);
		int x = leader(a), y = leader(b);
		if(x == y) return x;
		if(-parent_or_size[x] < -parent_or_size[y])
			std::swap(x, y);
		parent_or_size[x] += parent_or_size[y];
		parent_or_size[y] = x;
		return x;
	}

	bool same(int a, int b) {
		assert(0 <= a && a < _n);
		assert(0 <= b && b < _n);
		return leader(a) == leader(b);
	}

	int leader(int a) {
		assert(0 <= a && a < _n);
		if(parent_or_size[a] < 0) return a;
		return parent_or_size[a]
			= leader(parent_or_size[a]);
	}

	int size(int a) {
		assert(0 <= a && a < _n);
		return -parent_or_size[leader(a)];
	}

	std::vector<std::vector<int>> groups() {
		std::vector<int> leader_buf(_n), group_size(_n);
		for(int i = 0; i < _n; i++) {
			leader_buf[i] = leader(i);
			group_size[leader_buf[i]]++;
		}
		std::vector<std::vector<int>> result(_n);
		for(int i = 0; i < _n; i++) {
			result[i].reserve(group_size[i]);
		}
		for(int i = 0; i < _n; i++) {
			result[leader_buf[i]].push_back(i);
		}
		result.erase(
			std::remove_if(result.begin(), result.end(),
						   [&](const std::vector<int>& v) {
							   return v.empty();
						   }),
			result.end());
		return result;
	}

private:
	int _n;
	// root node: -1 * component size
	// otherwise: parent
	std::vector<int> parent_or_size;
};

}  // namespace atcoder


#include <cassert>
#include <vector>

namespace atcoder {

// Reference: https://en.wikipedia.org/wiki/Fenwick_tree
template<class T>
struct fenwick_tree {
	using U = internal::to_unsigned_t<T>;

public:
	fenwick_tree(): _n(0) {
	}
	fenwick_tree(int n): _n(n), data(n) {
	}

	void add(int p, T x) {
		assert(0 <= p && p < _n);
		p++;
		while(p <= _n) {
			data[p - 1] += U(x);
			p += p & -p;
		}
	}

	T sum(int l, int r) {
		assert(0 <= l && l <= r && r <= _n);
		return sum(r) - sum(l);
	}

	size_t sz() {
		return _n;
	}

private:
	int _n;
	std::vector<U> data;

	U sum(int r) {
		U s = 0;
		while(r > 0) {
			s += data[r - 1];
			r -= r & -r;
		}
		return s;
	}
};

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <iostream>
#include <vector>
namespace atcoder {

template<class S, S (*op)(S, S), S (*e)(), class F,
		 S (*mapping)(F, S), F (*composition)(F, F),
		 F (*id)()>
struct lazy_segtree {
public:
	lazy_segtree(): lazy_segtree(0) {
	}
	lazy_segtree(int n)
		: lazy_segtree(std::vector<S>(n, e())) {
	}
	lazy_segtree(const std::vector<S>& v)
		: _n(int(v.size())) {
		log = internal::ceil_pow2(_n);
		size = 1 << log;
		d = std::vector<S>(2 * size, e());
		lz = std::vector<F>(size, id());
		for(int i = 0; i < _n; i++) d[size + i] = v[i];
		for(int i = size - 1; i >= 1; i--) {
			update(i);
		}
	}

	void set(int p, S x) {
		assert(0 <= p && p < _n);
		p += size;
		for(int i = log; i >= 1; i--) push(p >> i);
		d[p] = x;
		for(int i = 1; i <= log; i++) update(p >> i);
	}

	S get(int p) {
		assert(0 <= p && p < _n);
		p += size;
		for(int i = log; i >= 1; i--) push(p >> i);
		return d[p];
	}

	S prod(int l, int r) {
		assert(0 <= l && l <= r && r <= _n);
		if(l == r) return e();

		l += size;
		r += size;

		for(int i = log; i >= 1; i--) {
			if(((l >> i) << i) != l) push(l >> i);
			if(((r >> i) << i) != r) push(r >> i);
		}

		S sml = e(), smr = e();
		while(l < r) {
			if(l & 1) sml = op(sml, d[l++]);
			if(r & 1) smr = op(d[--r], smr);
			l >>= 1;
			r >>= 1;
		}

		return op(sml, smr);
	}

	S all_prod() {
		return d[1];
	}

	void apply(int p, F f) {
		assert(0 <= p && p < _n);
		p += size;
		for(int i = log; i >= 1; i--) push(p >> i);
		d[p] = mapping(f, d[p]);
		for(int i = 1; i <= log; i++) update(p >> i);
	}
	void apply(int l, int r, F f) {
		assert(0 <= l && l <= r && r <= _n);
		if(l == r) return;

		l += size;
		r += size;

		for(int i = log; i >= 1; i--) {
			if(((l >> i) << i) != l) push(l >> i);
			if(((r >> i) << i) != r) push((r - 1) >> i);
		}

		{
			int l2 = l, r2 = r;
			while(l < r) {
				if(l & 1) all_apply(l++, f);
				if(r & 1) all_apply(--r, f);
				l >>= 1;
				r >>= 1;
			}
			l = l2;
			r = r2;
		}

		for(int i = 1; i <= log; i++) {
			if(((l >> i) << i) != l) update(l >> i);
			if(((r >> i) << i) != r) update((r - 1) >> i);
		}
	}

	template<bool (*g)(S)>
	int max_right(int l) {
		return max_right(l, [](S x) {
			return g(x);
		});
	}
	template<class G>
	int max_right(int l, G g) {
		assert(0 <= l && l <= _n);
		assert(g(e()));
		if(l == _n) return _n;
		l += size;
		for(int i = log; i >= 1; i--) push(l >> i);
		S sm = e();
		do {
			while(l % 2 == 0) l >>= 1;
			if(!g(op(sm, d[l]))) {
				while(l < size) {
					push(l);
					l = (2 * l);
					if(g(op(sm, d[l]))) {
						sm = op(sm, d[l]);
						l++;
					}
				}
				return l - size;
			}
			sm = op(sm, d[l]);
			l++;
		} while((l & -l) != l);
		return _n;
	}

	template<bool (*g)(S)>
	int min_left(int r) {
		return min_left(r, [](S x) {
			return g(x);
		});
	}
	template<class G>
	int min_left(int r, G g) {
		assert(0 <= r && r <= _n);
		assert(g(e()));
		if(r == 0) return 0;
		r += size;
		for(int i = log; i >= 1; i--) push((r - 1) >> i);
		S sm = e();
		do {
			r--;
			while(r > 1 && (r % 2)) r >>= 1;
			if(!g(op(d[r], sm))) {
				while(r < size) {
					push(r);
					r = (2 * r + 1);
					if(g(op(d[r], sm))) {
						sm = op(d[r], sm);
						r--;
					}
				}
				return r + 1 - size;
			}
			sm = op(d[r], sm);
		} while((r & -r) != r);
		return 0;
	}
	size_t sz() const {
		return _n;
	}

	std::vector<S> dump() {
		std::vector<S> ret;
		for(int i = 0; i < _n; i++) {
			ret.push_back(get(i));
		}
	}

private:
	int _n, size, log;
	std::vector<S> d;
	std::vector<F> lz;

	void update(int k) {
		d[k] = op(d[2 * k], d[2 * k + 1]);
	}
	void all_apply(int k, F f) {
		d[k] = mapping(f, d[k]);
		if(k < size) lz[k] = composition(f, lz[k]);
	}
	void push(int k) {
		all_apply(2 * k, lz[k]);
		all_apply(2 * k + 1, lz[k]);
		lz[k] = id();
	}
};

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <tuple>
#include <vector>

namespace atcoder {

long long pow_mod(long long x, long long n, int m) {
    assert(0 <= n && 1 <= m);
    if (m == 1) return 0;
    internal::barrett bt((unsigned int)(m));
    unsigned int r = 1, y = (unsigned int)(internal::safe_mod(x, m));
    while (n) {
        if (n & 1) r = bt.mul(r, y);
        y = bt.mul(y, y);
        n >>= 1;
    }
    return r;
}

long long inv_mod(long long x, long long m) {
    assert(1 <= m);
    auto z = internal::inv_gcd(x, m);
    assert(z.first == 1);
    return z.second;
}

// (rem, mod)
std::pair<long long, long long> crt(const std::vector<long long>& r,
                                    const std::vector<long long>& m) {
    assert(r.size() == m.size());
    int n = int(r.size());
    // Contracts: 0 <= r0 < m0
    long long r0 = 0, m0 = 1;
    for (int i = 0; i < n; i++) {
        assert(1 <= m[i]);
        long long r1 = internal::safe_mod(r[i], m[i]), m1 = m[i];
        if (m0 < m1) {
            std::swap(r0, r1);
            std::swap(m0, m1);
        }
        if (m0 % m1 == 0) {
            if (r0 % m1 != r1) return {0, 0};
            continue;
        }
        // assume: m0 > m1, lcm(m0, m1) >= 2 * max(m0, m1)

        // (r0, m0), (r1, m1) -> (r2, m2 = lcm(m0, m1));
        // r2 % m0 = r0
        // r2 % m1 = r1
        // -> (r0 + x*m0) % m1 = r1
        // -> x*u0*g % (u1*g) = (r1 - r0) (u0*g = m0, u1*g = m1)
        // -> x = (r1 - r0) / g * inv(u0) (mod u1)

        // im = inv(u0) (mod u1) (0 <= im < u1)
        long long g, im;
        std::tie(g, im) = internal::inv_gcd(m0, m1);

        long long u1 = (m1 / g);
        // |r1 - r0| < (m0 + m1) <= lcm(m0, m1)
        if ((r1 - r0) % g) return {0, 0};

        // u1 * u1 <= m1 * m1 / g / g <= m0 * m1 / g = lcm(m0, m1)
        long long x = (r1 - r0) / g % u1 * im % u1;

        // |r0| + |m0 * x|
        // < m0 + m0 * (u1 - 1)
        // = m0 + m0 * m1 / g - m0
        // = lcm(m0, m1)
        r0 += x * m0;
        m0 *= u1;  // -> lcm(m0, m1)
        if (r0 < 0) r0 += m0;
    }
    return {r0, m0};
}

long long floor_sum(long long n, long long m, long long a, long long b) {
    long long ans = 0;
    if (a >= m) {
        ans += (n - 1) * n * (a / m) / 2;
        a %= m;
    }
    if (b >= m) {
        ans += n * (b / m);
        b %= m;
    }

    long long y_max = (a * n + b) / m, x_max = (y_max * m - b);
    if (y_max == 0) return ans;
    ans += (n - (x_max + a - 1) / a) * y_max;
    ans += floor_sum(y_max, a, m, (a - x_max % a) % a);
    return ans;
}

}  // namespace atcoder


#include <algorithm>

#include <vector>

namespace atcoder {

namespace internal {

template <class T> struct simple_queue {
    std::vector<T> payload;
    int pos = 0;
    void reserve(int n) { payload.reserve(n); }
    int size() const { return int(payload.size()) - pos; }
    bool empty() const { return pos == int(payload.size()); }
    void push(const T& t) { payload.push_back(t); }
    T& front() { return payload[pos]; }
    void clear() {
        payload.clear();
        pos = 0;
    }
    void pop() { pos++; }
};

}  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <limits>
#include <queue>
#include <vector>

namespace atcoder {

template <class Cap> struct mf_graph {
  public:
    mf_graph() : _n(0) {}
    mf_graph(int n) : _n(n), g(n) {}

    int add_edge(int from, int to, Cap cap) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        assert(0 <= cap);
        int m = int(pos.size());
        pos.push_back({from, int(g[from].size())});
        int from_id = int(g[from].size());
        int to_id = int(g[to].size());
        if (from == to) to_id++;
        g[from].push_back(_edge{to, to_id, cap});
        g[to].push_back(_edge{from, from_id, 0});
        return m;
    }

    struct edge {
        int from, to;
        Cap cap, flow;
    };

    edge get_edge(int i) {
        int m = int(pos.size());
        assert(0 <= i && i < m);
        auto _e = g[pos[i].first][pos[i].second];
        auto _re = g[_e.to][_e.rev];
        return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
    }
    std::vector<edge> edges() {
        int m = int(pos.size());
        std::vector<edge> result;
        for (int i = 0; i < m; i++) {
            result.push_back(get_edge(i));
        }
        return result;
    }
    void change_edge(int i, Cap new_cap, Cap new_flow) {
        int m = int(pos.size());
        assert(0 <= i && i < m);
        assert(0 <= new_flow && new_flow <= new_cap);
        auto& _e = g[pos[i].first][pos[i].second];
        auto& _re = g[_e.to][_e.rev];
        _e.cap = new_cap - new_flow;
        _re.cap = new_flow;
    }

    Cap flow(int s, int t) {
        return flow(s, t, std::numeric_limits<Cap>::max());
    }
    Cap flow(int s, int t, Cap flow_limit) {
        assert(0 <= s && s < _n);
        assert(0 <= t && t < _n);
        assert(s != t);

        std::vector<int> level(_n), iter(_n);
        internal::simple_queue<int> que;

        auto bfs = [&]() {
            std::fill(level.begin(), level.end(), -1);
            level[s] = 0;
            que.clear();
            que.push(s);
            while (!que.empty()) {
                int v = que.front();
                que.pop();
                for (auto e : g[v]) {
                    if (e.cap == 0 || level[e.to] >= 0) continue;
                    level[e.to] = level[v] + 1;
                    if (e.to == t) return;
                    que.push(e.to);
                }
            }
        };
        auto dfs = [&](auto self, int v, Cap up) {
            if (v == s) return up;
            Cap res = 0;
            int level_v = level[v];
            for (int& i = iter[v]; i < int(g[v].size()); i++) {
                _edge& e = g[v][i];
                if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0) continue;
                Cap d =
                    self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));
                if (d <= 0) continue;
                g[v][i].cap += d;
                g[e.to][e.rev].cap -= d;
                res += d;
                if (res == up) break;
            }
            return res;
        };

        Cap flow = 0;
        while (flow < flow_limit) {
            bfs();
            if (level[t] == -1) break;
            std::fill(iter.begin(), iter.end(), 0);
            while (flow < flow_limit) {
                Cap f = dfs(dfs, t, flow_limit - flow);
                if (!f) break;
                flow += f;
            }
        }
        return flow;
    }

    std::vector<bool> min_cut(int s) {
        std::vector<bool> visited(_n);
        internal::simple_queue<int> que;
        que.push(s);
        while (!que.empty()) {
            int p = que.front();
            que.pop();
            visited[p] = true;
            for (auto e : g[p]) {
                if (e.cap && !visited[e.to]) {
                    visited[e.to] = true;
                    que.push(e.to);
                }
            }
        }
        return visited;
    }

  private:
    int _n;
    struct _edge {
        int to, rev;
        Cap cap;
    };
    std::vector<std::pair<int, int>> pos;
    std::vector<std::vector<_edge>> g;
};

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <limits>
#include <queue>
#include <vector>

namespace atcoder {

template <class Cap, class Cost> struct mcf_graph {
  public:
    mcf_graph() {}
    mcf_graph(int n) : _n(n), g(n) {}

    int add_edge(int from, int to, Cap cap, Cost cost) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        int m = int(pos.size());
        pos.push_back({from, int(g[from].size())});
        int from_id = int(g[from].size());
        int to_id = int(g[to].size());
        if (from == to) to_id++;
        g[from].push_back(_edge{to, to_id, cap, cost});
        g[to].push_back(_edge{from, from_id, 0, -cost});
        return m;
    }

    struct edge {
        int from, to;
        Cap cap, flow;
        Cost cost;
    };

    edge get_edge(int i) {
        int m = int(pos.size());
        assert(0 <= i && i < m);
        auto _e = g[pos[i].first][pos[i].second];
        auto _re = g[_e.to][_e.rev];
        return edge{
            pos[i].first, _e.to, _e.cap + _re.cap, _re.cap, _e.cost,
        };
    }
    std::vector<edge> edges() {
        int m = int(pos.size());
        std::vector<edge> result(m);
        for (int i = 0; i < m; i++) {
            result[i] = get_edge(i);
        }
        return result;
    }

    std::pair<Cap, Cost> flow(int s, int t) {
        return flow(s, t, std::numeric_limits<Cap>::max());
    }
    std::pair<Cap, Cost> flow(int s, int t, Cap flow_limit) {
        return slope(s, t, flow_limit).back();
    }
    std::vector<std::pair<Cap, Cost>> slope(int s, int t) {
        return slope(s, t, std::numeric_limits<Cap>::max());
    }
    std::vector<std::pair<Cap, Cost>> slope(int s, int t, Cap flow_limit) {
        assert(0 <= s && s < _n);
        assert(0 <= t && t < _n);
        assert(s != t);
        // variants (C = maxcost):
        // -(n-1)C <= dual[s] <= dual[i] <= dual[t] = 0
        // reduced cost (= e.cost + dual[e.from] - dual[e.to]) >= 0 for all edge
        std::vector<Cost> dual(_n, 0), dist(_n);
        std::vector<int> pv(_n), pe(_n);
        std::vector<bool> vis(_n);
        auto dual_ref = [&]() {
            std::fill(dist.begin(), dist.end(),
                      std::numeric_limits<Cost>::max());
            std::fill(pv.begin(), pv.end(), -1);
            std::fill(pe.begin(), pe.end(), -1);
            std::fill(vis.begin(), vis.end(), false);
            struct Q {
                Cost key;
                int to;
                bool operator<(Q r) const { return key > r.key; }
            };
            std::priority_queue<Q> que;
            dist[s] = 0;
            que.push(Q{0, s});
            while (!que.empty()) {
                int v = que.top().to;
                que.pop();
                if (vis[v]) continue;
                vis[v] = true;
                if (v == t) break;
                // dist[v] = shortest(s, v) + dual[s] - dual[v]
                // dist[v] >= 0 (all reduced cost are positive)
                // dist[v] <= (n-1)C
                for (int i = 0; i < int(g[v].size()); i++) {
                    auto e = g[v][i];
                    if (vis[e.to] || !e.cap) continue;
                    // |-dual[e.to] + dual[v]| <= (n-1)C
                    // cost <= C - -(n-1)C + 0 = nC
                    Cost cost = e.cost - dual[e.to] + dual[v];
                    if (dist[e.to] - dist[v] > cost) {
                        dist[e.to] = dist[v] + cost;
                        pv[e.to] = v;
                        pe[e.to] = i;
                        que.push(Q{dist[e.to], e.to});
                    }
                }
            }
            if (!vis[t]) {
                return false;
            }

            for (int v = 0; v < _n; v++) {
                if (!vis[v]) continue;
                // dual[v] = dual[v] - dist[t] + dist[v]
                //         = dual[v] - (shortest(s, t) + dual[s] - dual[t]) + (shortest(s, v) + dual[s] - dual[v])
                //         = - shortest(s, t) + dual[t] + shortest(s, v)
                //         = shortest(s, v) - shortest(s, t) >= 0 - (n-1)C
                dual[v] -= dist[t] - dist[v];
            }
            return true;
        };
        Cap flow = 0;
        Cost cost = 0, prev_cost_per_flow = -1;
        std::vector<std::pair<Cap, Cost>> result;
        result.push_back({flow, cost});
        while (flow < flow_limit) {
            if (!dual_ref()) break;
            Cap c = flow_limit - flow;
            for (int v = t; v != s; v = pv[v]) {
                c = std::min(c, g[pv[v]][pe[v]].cap);
            }
            for (int v = t; v != s; v = pv[v]) {
                auto& e = g[pv[v]][pe[v]];
                e.cap -= c;
                g[v][e.rev].cap += c;
            }
            Cost d = -dual[s];
            flow += c;
            cost += c * d;
            if (prev_cost_per_flow == d) {
                result.pop_back();
            }
            result.push_back({flow, cost});
            prev_cost_per_flow = d;
        }
        return result;
    }

  private:
    int _n;

    struct _edge {
        int to, rev;
        Cap cap;
        Cost cost;
    };

    std::vector<std::pair<int, int>> pos;
    std::vector<std::vector<_edge>> g;
};

}  // namespace atcoder


#include <algorithm>

#include <algorithm>
#include <utility>
#include <vector>

namespace atcoder {
namespace internal {

template <class E> struct csr {
    std::vector<int> start;
    std::vector<E> elist;
    csr(int n, const std::vector<std::pair<int, E>>& edges)
        : start(n + 1), elist(edges.size()) {
        for (auto e : edges) {
            start[e.first + 1]++;
        }
        for (int i = 1; i <= n; i++) {
            start[i] += start[i - 1];
        }
        auto counter = start;
        for (auto e : edges) {
            elist[counter[e.first]++] = e.second;
        }
    }
};

// Reference:
// R. Tarjan,
// Depth-First Search and Linear Graph Algorithms
struct scc_graph {
  public:
    scc_graph(int n) : _n(n) {}

    int num_vertices() { return _n; }

    void add_edge(int from, int to) { edges.push_back({from, {to}}); }

    // @return pair of (# of scc, scc id)
    std::pair<int, std::vector<int>> scc_ids() {
        auto g = csr<edge>(_n, edges);
        int now_ord = 0, group_num = 0;
        std::vector<int> visited, low(_n), ord(_n, -1), ids(_n);
        visited.reserve(_n);
        auto dfs = [&](auto self, int v) -> void {
            low[v] = ord[v] = now_ord++;
            visited.push_back(v);
            for (int i = g.start[v]; i < g.start[v + 1]; i++) {
                auto to = g.elist[i].to;
                if (ord[to] == -1) {
                    self(self, to);
                    low[v] = std::min(low[v], low[to]);
                } else {
                    low[v] = std::min(low[v], ord[to]);
                }
            }
            if (low[v] == ord[v]) {
                while (true) {
                    int u = visited.back();
                    visited.pop_back();
                    ord[u] = _n;
                    ids[u] = group_num;
                    if (u == v) break;
                }
                group_num++;
            }
        };
        for (int i = 0; i < _n; i++) {
            if (ord[i] == -1) dfs(dfs, i);
        }
        for (auto& x : ids) {
            x = group_num - 1 - x;
        }
        return {group_num, ids};
    }

    std::vector<std::vector<int>> scc() {
        auto ids = scc_ids();
        int group_num = ids.first;
        std::vector<int> counts(group_num);
        for (auto x : ids.second) counts[x]++;
        std::vector<std::vector<int>> groups(ids.first);
        for (int i = 0; i < group_num; i++) {
            groups[i].reserve(counts[i]);
        }
        for (int i = 0; i < _n; i++) {
            groups[ids.second[i]].push_back(i);
        }
        return groups;
    }

  private:
    int _n;
    struct edge {
        int to;
    };
    std::vector<std::pair<int, edge>> edges;
};

}  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <vector>

namespace atcoder {

struct scc_graph {
  public:
    scc_graph() : internal(0) {}
    scc_graph(int n) : internal(n) {}

    void add_edge(int from, int to) {
        int n = internal.num_vertices();
        assert(0 <= from && from < n);
        assert(0 <= to && to < n);
        internal.add_edge(from, to);
    }

    std::vector<std::vector<int>> scc() { return internal.scc(); }

  private:
    internal::scc_graph internal;
};

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <vector>

namespace atcoder {

template<class S, S (*op)(S, S), S (*e)()>
struct segtree {
public:
	segtree(): segtree(0) {
	}
	segtree(int n): segtree(std::vector<S>(n, e())) {
	}
	segtree(const std::vector<S>& v): _n(int(v.size())) {
		log = internal::ceil_pow2(_n);
		sz = 1 << log;
		d = std::vector<S>(2 * sz, e());
		for(int i = 0; i < _n; i++) d[sz + i] = v[i];
		for(int i = sz - 1; i >= 1; i--) {
			update(i);
		}
	}

	void set(int p, S x) {
		assert(0 <= p && p < _n);
		p += sz;
		d[p] = x;
		for(int i = 1; i <= log; i++) update(p >> i);
	}

	S get(int p) const {
		assert(0 <= p && p < _n);
		return d[p + sz];
	}

	S prod(int l, int r) {
		assert(0 <= l && l <= r && r <= _n);
		S sml = e(), smr = e();
		l += sz;
		r += sz;

		while(l < r) {
			if(l & 1) sml = op(sml, d[l++]);
			if(r & 1) smr = op(d[--r], smr);
			l >>= 1;
			r >>= 1;
		}
		return op(sml, smr);
	}

	S all_prod() {
		return d[1];
	}

	template<bool (*f)(S)>
	int max_right(int l) {
		return max_right(l, [](S x) {
			return f(x);
		});
	}
	template<class F>
	int max_right(int l, F f) {
		assert(0 <= l && l <= _n);
		assert(f(e()));
		if(l == _n) return _n;
		l += sz;
		S sm = e();
		do {
			while(l % 2 == 0) l >>= 1;
			if(!f(op(sm, d[l]))) {
				while(l < sz) {
					l = (2 * l);
					if(f(op(sm, d[l]))) {
						sm = op(sm, d[l]);
						l++;
					}
				}
				return l - sz;
			}
			sm = op(sm, d[l]);
			l++;
		} while((l & -l) != l);
		return _n;
	}

	template<bool (*f)(S)>
	int min_left(int r) {
		return min_left(r, [](S x) {
			return f(x);
		});
	}
	template<class F>
	int min_left(int r, F f) {
		assert(0 <= r && r <= _n);
		assert(f(e()));
		if(r == 0) return 0;
		r += sz;
		S sm = e();
		do {
			r--;
			while(r > 1 && (r % 2)) r >>= 1;
			if(!f(op(d[r], sm))) {
				while(r < sz) {
					r = (2 * r + 1);
					if(f(op(d[r], sm))) {
						sm = op(d[r], sm);
						r--;
					}
				}
				return r + 1 - sz;
			}
			sm = op(d[r], sm);
		} while((r & -r) != r);
		return 0;
	}
	size_t size() const {
		return _n;
	}

private:
	int _n, sz, log;
	std::vector<S> d;

	void update(int k) {
		d[k] = op(d[2 * k], d[2 * k + 1]);
	}
};

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <numeric>
#include <string>
#include <vector>

namespace atcoder {

namespace internal {

std::vector<int> sa_naive(const std::vector<int>& s) {
    int n = int(s.size());
    std::vector<int> sa(n);
    std::iota(sa.begin(), sa.end(), 0);
    std::sort(sa.begin(), sa.end(), [&](int l, int r) {
        if (l == r) return false;
        while (l < n && r < n) {
            if (s[l] != s[r]) return s[l] < s[r];
            l++;
            r++;
        }
        return l == n;
    });
    return sa;
}

std::vector<int> sa_doubling(const std::vector<int>& s) {
    int n = int(s.size());
    std::vector<int> sa(n), rnk = s, tmp(n);
    std::iota(sa.begin(), sa.end(), 0);
    for (int k = 1; k < n; k *= 2) {
        auto cmp = [&](int x, int y) {
            if (rnk[x] != rnk[y]) return rnk[x] < rnk[y];
            int rx = x + k < n ? rnk[x + k] : -1;
            int ry = y + k < n ? rnk[y + k] : -1;
            return rx < ry;
        };
        std::sort(sa.begin(), sa.end(), cmp);
        tmp[sa[0]] = 0;
        for (int i = 1; i < n; i++) {
            tmp[sa[i]] = tmp[sa[i - 1]] + (cmp(sa[i - 1], sa[i]) ? 1 : 0);
        }
        std::swap(tmp, rnk);
    }
    return sa;
}

// SA-IS, linear-time suffix array construction
// Reference:
// G. Nong, S. Zhang, and W. H. Chan,
// Two Efficient Algorithms for Linear Time Suffix Array Construction
template <int THRESHOLD_NAIVE = 10, int THRESHOLD_DOUBLING = 40>
std::vector<int> sa_is(const std::vector<int>& s, int upper) {
    int n = int(s.size());
    if (n == 0) return {};
    if (n == 1) return {0};
    if (n == 2) {
        if (s[0] < s[1]) {
            return {0, 1};
        } else {
            return {1, 0};
        }
    }
    if (n < THRESHOLD_NAIVE) {
        return sa_naive(s);
    }
    if (n < THRESHOLD_DOUBLING) {
        return sa_doubling(s);
    }

    std::vector<int> sa(n);
    std::vector<bool> ls(n);
    for (int i = n - 2; i >= 0; i--) {
        ls[i] = (s[i] == s[i + 1]) ? ls[i + 1] : (s[i] < s[i + 1]);
    }
    std::vector<int> sum_l(upper + 1), sum_s(upper + 1);
    for (int i = 0; i < n; i++) {
        if (!ls[i]) {
            sum_s[s[i]]++;
        } else {
            sum_l[s[i] + 1]++;
        }
    }
    for (int i = 0; i <= upper; i++) {
        sum_s[i] += sum_l[i];
        if (i < upper) sum_l[i + 1] += sum_s[i];
    }

    auto induce = [&](const std::vector<int>& lms) {
        std::fill(sa.begin(), sa.end(), -1);
        std::vector<int> buf(upper + 1);
        std::copy(sum_s.begin(), sum_s.end(), buf.begin());
        for (auto d : lms) {
            if (d == n) continue;
            sa[buf[s[d]]++] = d;
        }
        std::copy(sum_l.begin(), sum_l.end(), buf.begin());
        sa[buf[s[n - 1]]++] = n - 1;
        for (int i = 0; i < n; i++) {
            int v = sa[i];
            if (v >= 1 && !ls[v - 1]) {
                sa[buf[s[v - 1]]++] = v - 1;
            }
        }
        std::copy(sum_l.begin(), sum_l.end(), buf.begin());
        for (int i = n - 1; i >= 0; i--) {
            int v = sa[i];
            if (v >= 1 && ls[v - 1]) {
                sa[--buf[s[v - 1] + 1]] = v - 1;
            }
        }
    };

    std::vector<int> lms_map(n + 1, -1);
    int m = 0;
    for (int i = 1; i < n; i++) {
        if (!ls[i - 1] && ls[i]) {
            lms_map[i] = m++;
        }
    }
    std::vector<int> lms;
    lms.reserve(m);
    for (int i = 1; i < n; i++) {
        if (!ls[i - 1] && ls[i]) {
            lms.push_back(i);
        }
    }

    induce(lms);

    if (m) {
        std::vector<int> sorted_lms;
        sorted_lms.reserve(m);
        for (int v : sa) {
            if (lms_map[v] != -1) sorted_lms.push_back(v);
        }
        std::vector<int> rec_s(m);
        int rec_upper = 0;
        rec_s[lms_map[sorted_lms[0]]] = 0;
        for (int i = 1; i < m; i++) {
            int l = sorted_lms[i - 1], r = sorted_lms[i];
            int end_l = (lms_map[l] + 1 < m) ? lms[lms_map[l] + 1] : n;
            int end_r = (lms_map[r] + 1 < m) ? lms[lms_map[r] + 1] : n;
            bool same = true;
            if (end_l - l != end_r - r) {
                same = false;
            } else {
                while (l < end_l) {
                    if (s[l] != s[r]) {
                        break;
                    }
                    l++;
                    r++;
                }
                if (l == n || s[l] != s[r]) same = false;
            }
            if (!same) rec_upper++;
            rec_s[lms_map[sorted_lms[i]]] = rec_upper;
        }

        auto rec_sa =
            sa_is<THRESHOLD_NAIVE, THRESHOLD_DOUBLING>(rec_s, rec_upper);

        for (int i = 0; i < m; i++) {
            sorted_lms[i] = lms[rec_sa[i]];
        }
        induce(sorted_lms);
    }
    return sa;
}

}  // namespace internal

std::vector<int> suffix_array(const std::vector<int>& s, int upper) {
    assert(0 <= upper);
    for (int d : s) {
        assert(0 <= d && d <= upper);
    }
    auto sa = internal::sa_is(s, upper);
    return sa;
}

template <class T> std::vector<int> suffix_array(const std::vector<T>& s) {
    int n = int(s.size());
    std::vector<int> idx(n);
    iota(idx.begin(), idx.end(), 0);
    sort(idx.begin(), idx.end(), [&](int l, int r) { return s[l] < s[r]; });
    std::vector<int> s2(n);
    int now = 0;
    for (int i = 0; i < n; i++) {
        if (i && s[idx[i - 1]] != s[idx[i]]) now++;
        s2[idx[i]] = now;
    }
    return internal::sa_is(s2, now);
}

std::vector<int> suffix_array(const std::string& s) {
    int n = int(s.size());
    std::vector<int> s2(n);
    for (int i = 0; i < n; i++) {
        s2[i] = s[i];
    }
    return internal::sa_is(s2, 255);
}

// Reference:
// T. Kasai, G. Lee, H. Arimura, S. Arikawa, and K. Park,
// Linear-Time Longest-Common-Prefix Computation in Suffix Arrays and Its
// Applications
template <class T>
std::vector<int> lcp_array(const std::vector<T>& s,
                           const std::vector<int>& sa) {
    int n = int(s.size());
    assert(n >= 1);
    std::vector<int> rnk(n);
    for (int i = 0; i < n; i++) {
        rnk[sa[i]] = i;
    }
    std::vector<int> lcp(n - 1);
    int h = 0;
    for (int i = 0; i < n; i++) {
        if (h > 0) h--;
        if (rnk[i] == 0) continue;
        int j = sa[rnk[i] - 1];
        for (; j + h < n && i + h < n; h++) {
            if (s[j + h] != s[i + h]) break;
        }
        lcp[rnk[i] - 1] = h;
    }
    return lcp;
}

std::vector<int> lcp_array(const std::string& s, const std::vector<int>& sa) {
    int n = int(s.size());
    std::vector<int> s2(n);
    for (int i = 0; i < n; i++) {
        s2[i] = s[i];
    }
    return lcp_array(s2, sa);
}

// Reference:
// D. Gusfield,
// Algorithms on Strings, Trees, and Sequences: Computer Science and
// Computational Biology
template <class T> std::vector<int> z_algorithm(const std::vector<T>& s) {
    int n = int(s.size());
    if (n == 0) return {};
    std::vector<int> z(n);
    z[0] = 0;
    for (int i = 1, j = 0; i < n; i++) {
        int& k = z[i];
        k = (j + z[j] <= i) ? 0 : std::min(j + z[j] - i, z[i - j]);
        while (i + k < n && s[k] == s[i + k]) k++;
        if (j + z[j] < i + z[i]) j = i;
    }
    z[0] = n;
    return z;
}

std::vector<int> z_algorithm(const std::string& s) {
    int n = int(s.size());
    std::vector<int> s2(n);
    for (int i = 0; i < n; i++) {
        s2[i] = s[i];
    }
    return z_algorithm(s2);
}

}  // namespace atcoder


#include <cassert>
#include <vector>

namespace atcoder {

// Reference:
// B. Aspvall, M. Plass, and R. Tarjan,
// A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean
// Formulas
struct two_sat {
  public:
    two_sat() : _n(0), scc(0) {}
    two_sat(int n) : _n(n), _answer(n), scc(2 * n) {}

    void add_clause(int i, bool f, int j, bool g) {
        assert(0 <= i && i < _n);
        assert(0 <= j && j < _n);
        scc.add_edge(2 * i + (f ? 0 : 1), 2 * j + (g ? 1 : 0));
        scc.add_edge(2 * j + (g ? 0 : 1), 2 * i + (f ? 1 : 0));
    }
    bool satisfiable() {
        auto id = scc.scc_ids().second;
        for (int i = 0; i < _n; i++) {
            if (id[2 * i] == id[2 * i + 1]) return false;
            _answer[i] = id[2 * i] < id[2 * i + 1];
        }
        return true;
    }
    std::vector<bool> answer() { return _answer; }

  private:
    int _n;
    std::vector<bool> _answer;
    internal::scc_graph scc;
};

}  // namespace atcoder

using namespace atcoder;
#include <bits/stdc++.h>

using namespace std;
#define SZ(x) (int) (x).size()
#define REP(i, n) for(int i = 0; i < (n); i++)
#define FOR(i, a, b) for(auto i = (a); i < (b); i++)
#define For(i, a, b, c) \
	for(auto i = (a); i != (b); i += (c))
#define REPR(i, n) for(auto i = (n) - 1; i >= 0; i--)
#define ALL(s) (s).begin(), (s).end()
#define so(V) sort(ALL(V))
#define rev(V) reverse(ALL(V))
#define uni(v) v.erase(unique(ALL(v)), (v).end())
#define eb emplace_back

typedef unsigned long long ull;
typedef long long ll;
typedef vector<int> vi;
typedef vector<ll> vll;
typedef vector<bool> vb;
typedef vector<vi> vvi;
typedef vector<vll> vvll;
typedef pair<int, int> PI;
typedef pair<ll, ll> PL;
const double EPS = 1e-6;
const int MOD = 1e9 + 7;
const int INF = (1 << 30);
const ll LINF = 1e18;
const long double math_PI = acosl(-1);

template<typename T>
vector<T> make_v(size_t a) {
	return vector<T>(a);
}

template<typename T, typename... Ts>
auto make_v(size_t a, Ts... ts) {
	return vector<decltype(make_v<T>(ts...))>(
		a, make_v<T>(ts...));
}

template<typename T, typename V>
typename enable_if<is_class<T>::value == 0>::type fill_v(
	T &t, const V &v) {
	t = v;
}

template<typename T, typename V>
typename enable_if<is_class<T>::value != 0>::type fill_v(
	T &t, const V &v) {
	for(auto &e: t) fill_v(e, v);
}

template<class T>
bool chmax(T &a, const T &b) {
	if(a < b) {
		a = b;
		return true;
	}
	return false;
}

template<class T>
bool chmin(T &a, const T &b) {
	if(a > b) {
		a = b;
		return true;
	}
	return false;
}

template<typename S, typename T>
istream &operator>>(istream &is, pair<S, T> &p) {
	cin >> p.first >> p.second;
	return is;
}

template<typename T>
istream &operator>>(istream &is, vector<T> &vec) {
	for(T &x: vec) is >> x;
	return is;
}

template<typename T>
string join(vector<T> &vec, string splitter) {
	stringstream ss;
	REP(i, SZ(vec)) {
		if(i != 0) ss << splitter;
		ss << vec[i];
	}
	return ss.str();
}

template<typename T>
ostream &operator<<(ostream &os, vector<T> &vec) {
	os << join(vec, " ");
	return os;
}

#ifdef LOCAL
#include "cpp-dump/cpp-dump.hpp"
CPP_DUMP_DEFINE_EXPORT_OBJECT_GENERIC(dump())
#define dump(...) cpp_dump(__VA_ARGS__)
namespace cp = cpp_dump;
#else
#define dump(...)
#define preprocess(...)
#define CPP_DUMP_SET_OPTION(...)
#define CPP_DUMP_DEFINE_EXPORT_OBJECT(...)
#define CPP_DUMP_DEFINE_EXPORT_ENUM(...)
#define CPP_DUMP_DEFINE_DANGEROUS_EXPORT_OBJECT(...)
#endif
template<typename T>
struct Comb {
private:
	vector<T> _fact;
	vector<T> inv;
	int pre_build;
	void expand(int x) {
		if(pre_build == -1) return;
		while(SZ(_fact) <= x) {
			int next = SZ(_fact);
			_fact.push_back(_fact[next - 1] * next);
			inv.push_back(_fact[next].inv());
		}
	}

	T nPr_calc(int n, int r) {
		T ret = 1;
		for(int i = n; i > n - r; i--) ret *= i;
		return ret;
	}
	T nCr_calc(int n, int r) {
		T ret = 1;
		REP(i, r) {
			ret *= n - i;
			ret /= i + 1;
		}
		return ret;
	}

public:
	Comb(int pre_build = 0): pre_build(pre_build) {
		if(pre_build == -1) {
			return;
		}
		_fact = vector<T>(pre_build + 1);
		_fact[0] = 1;
		inv = vector<T>(pre_build + 1);
		REP(i, pre_build) {
			_fact[i + 1] = _fact[i] * (i + 1);
		}
		inv[pre_build] = _fact[pre_build].inv();
		REPR(i, pre_build) {
			inv[i] = inv[i + 1] * (i + 1);
		}
	}
	T P(int n, int r) {
		assert(r >= 0);
		assert(n >= r);
		expand(n);
		if(pre_build == -1) return nPr_calc(n, r);
		return _fact[n] * inv[n - r];
	}
	T C(int n, int r) {
		assert(r >= 0);
		assert(n >= r);
		expand(n);
		if(pre_build == -1) return nCr_calc(n, r);
		return _fact[n] * inv[r] * inv[n - r];
	}
	T H(int n, int r) {
		assert(r >= 0);
		assert(n >= 1);
		expand(n + r - 1);
		return C(n + r - 1, r);
	}
	T type_permission(vi V) {
		assert(pre_build != -1);
		int sum = 0;
		for(int i: V) {
			assert(i >= 0);
			sum += i;
		}
		expand(sum);
		T ans = _fact[sum];
		for(int i: V) ans *= inv[i];
		return ans;
	}
	T fact(int n) {
		expand(n);
		return _fact[n];
	}
	T inv_fact(int n) {
		expand(n);
		return inv[n];
	}
};

template<typename T>
struct RangeSum {
private:
	vector<T> V;
	int N = -1;

public:
	RangeSum(vector<T> &v) {
		N = SZ(v);
		V = vector<T>(N + 1);
		V[0] = T(0);
		REP(i, N) {
			V[i + 1] = v[i] + V[i];
		}
	}

	T sum(int l, int r) {
		chmax(l, 0);
		chmin(r, N);
		chmax(r, l);
		return V[r] - V[l];
	}
};

template<typename T>
struct RangeSum2D {
private:
	vector<vector<T>> V;
	int H = -1;
	int W = -1;

public:
	RangeSum2D(vector<vector<T>> &v) {
		H = SZ(v);
		W = SZ(v[0]);
		V = vector<vector<T>>(H, vector<T>(W));
		REP(i, H) {
			REP(j, W) {
				V[i][j] += v[i][j];
				if(i != 0) V[i][j] += V[i - 1][j];
				if(j != 0) V[i][j] += V[i][j - 1];
				if(i != 0 && j != 0)
					V[i][j] -= V[i - 1][j - 1];
			}
		}
	}

	T sum(int y1, int x1, int y2, int x2) {
		T ret = V[y2][x2];
		if(y1 != 0) ret -= V[y1 - 1][x2];
		if(x1 != 0) ret -= V[y2][x1 - 1];
		if(y1 != 0 && x1 != 0) ret += V[y1 - 1][x1 - 1];
		return ret;
	}
};
using vi = vector<int>;
using ll = long long;
using mint = modint;

int main() {
	int T;
	cin >> T;
	while(T--) {
		int N, P;
		cin >> N >> P;
		int b0, A, B;
		cin >> b0 >> A >> B;
		mint::set_mod(P);
		vector<mint> V(2 * N);
		V[0] = 0;
		Comb<mint> comb(2 * N);
		for(int i = 1; i < 2 * N; i += 2) {
			int inner = (i - 1) / 2;
			V[i] = comb.C(2 * inner, inner) / (inner + 1);
			int outer = (2 * N - inner * 2 - 2) / 2;
			V[i] *= comb.C(2 * outer, outer) / (outer + 1);
		}
		RangeSum<mint> R(V);
		mint b = A * b0 + B;
		mint a = b + 1;
		mint ans = 0;
		REP(i, 2 * N) {
			int right = 2 * N - i - 1;
			int left = i;
			ans -= R.sum(0, right + 1) * a;
			ans += R.sum(0, left + 1) * a;
			b = A * b + B;
			a += b + 1;
		}
		cout << (ans / comb.C(2 * N, N) * (N + 1)).val()
			 << endl;
	}
}