#include <bits/stdc++.h>
#ifdef LOCAL
#include <debug.hpp>
#else
#define debug(...) void(0)
#endif

template <class T> std::istream& operator>>(std::istream& is, std::vector<T>& v) {
    for (auto& e : v) {
        is >> e;
    }
    return is;
}

template <class T> std::ostream& operator<<(std::ostream& os, const std::vector<T>& v) {
    for (std::string_view sep = ""; const auto& e : v) {
        os << std::exchange(sep, " ") << e;
    }
    return os;
}

template <class T, class U = T> bool chmin(T& x, U&& y) {
    return y < x and (x = std::forward<U>(y), true);
}

template <class T, class U = T> bool chmax(T& x, U&& y) {
    return x < y and (x = std::forward<U>(y), true);
}

template <class T> void mkuni(std::vector<T>& v) {
    std::ranges::sort(v);
    auto result = std::ranges::unique(v);
    v.erase(result.begin(), result.end());
}

template <class T> int lwb(const std::vector<T>& v, const T& x) {
    return std::distance(v.begin(), std::ranges::lower_bound(v, x));
}

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

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`
    explicit 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);

// @param n `n < 2^32`
// @param m `1 <= m < 2^32`
// @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64)
unsigned long long floor_sum_unsigned(unsigned long long n,
                                      unsigned long long m,
                                      unsigned long long a,
                                      unsigned long long b) {
    unsigned long long ans = 0;
    while (true) {
        if (a >= m) {
            ans += n * (n - 1) / 2 * (a / m);
            a %= m;
        }
        if (b >= m) {
            ans += n * (b / m);
            b %= m;
        }

        unsigned long long y_max = a * n + b;
        if (y_max < m) break;
        // y_max < m * (n + 1)
        // floor(y_max / m) <= n
        n = (unsigned long long)(y_max / m);
        b = (unsigned long long)(y_max % m);
        std::swap(m, a);
    }
    return ans;
}

}  // namespace internal

}  // namespace atcoder

#include <type_traits>

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

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

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());
    }

    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;
    }

  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());
    }

    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;
    }

  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

template <typename T> struct Matrix {
    std::vector<std::vector<T>> A;

    Matrix() = default;

    Matrix(int n, int m) : A(n, std::vector<T>(m, 0)) {}

    Matrix(int n) : A(n, std::vector<T>(n, 0)) {}

    bool empty() const { return A.empty(); }

    int size() const { return A.size(); }

    int height() const { return A.size(); }

    int width() const {
        assert(not A.empty());
        return A[0].size();
    }

    inline const std::vector<T>& operator[](int i) const { return A[i]; }

    inline std::vector<T>& operator[](int i) { return A[i]; }

    static Matrix identity(int n) {
        Matrix res(n);
        for (int i = 0; i < n; i++) res[i][i] = 1;
        return res;
    }

    Matrix& operator+=(const Matrix& B) {
        int n = height(), m = width();
        assert(n == B.height() and m == B.width());
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                (*this)[i][j] += B[i][j];
            }
        }
        return *this;
    }

    Matrix& operator-=(const Matrix& B) {
        int n = height(), m = width();
        assert(n == B.height() and m == B.width());
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                (*this)[i][j] -= B[i][j];
            }
        }
        return *this;
    }

    Matrix& operator*=(const Matrix& B) {
        int n = height(), m = B.width(), p = width();
        assert(p == B.height());
        std::vector<std::vector<T>> C(n, std::vector<T>(m, 0));
        for (int i = 0; i < n; i++) {
            for (int k = 0; k < p; k++) {
                for (int j = 0; j < m; j++) {
                    C[i][j] += (*this)[i][k] * B[k][j];
                }
            }
        }
        std::swap(A, C);
        return *this;
    }

    Matrix& operator*=(const T& v) {
        for (int i = 0; i < height(); i++) {
            for (int j = 0; j < width(); j++) {
                (*this)[i][j] *= v;
            }
        }
        return *this;
    }

    Matrix& operator/=(const T& v) {
        T inv = T(1) / v;
        for (int i = 0; i < height(); i++) {
            for (int j = 0; j < width(); j++) {
                (*this)[i][j] *= inv;
            }
        }
        return *this;
    }

    Matrix operator-() const {
        Matrix res(height(), width());
        for (int i = 0; i < height(); i++) {
            for (int j = 0; j < width(); j++) {
                res[i][j] = -(*this)[i][j];
            }
        }
        return res;
    }

    Matrix operator+(const Matrix& B) const { return Matrix(*this) += B; }

    Matrix operator-(const Matrix& B) const { return Matrix(*this) -= B; }

    Matrix operator*(const Matrix& B) const { return Matrix(*this) *= B; }

    Matrix operator*(const T& v) const { return Matrix(*this) *= v; }

    Matrix operator/(const T& v) const { return Matrix(*this) /= v; }

    bool operator==(const Matrix& B) const {
        assert(height() == B.height() && width() == B.width());
        return A == B.A;
    }

    bool operator!=(const Matrix& B) const {
        assert(height() == B.height() && width() == B.width());
        return A != B.A;
    }

    Matrix pow(long long n) const {
        assert(0 <= n);
        Matrix x = *this, r = identity(size());
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }

    Matrix transpose() const {
        Matrix res(width(), height());
        for (int i = 0; i < height(); i++) {
            for (int j = 0; j < width(); j++) {
                res[j][i] = (*this)[i][j];
            }
        }
        return res;
    }

    int rank() const { return Matrix(*this).gauss_jordan().first; }

    T det() const { return Matrix(*this).gauss_jordan().second; }

    Matrix inv() const {
        assert(height() == width());
        int n = height();
        Matrix B(*this);
        for (int i = 0; i < n; i++) {
            B[i].resize(2 * n, T(0));
            B[i][n + i] = T(1);
        }
        int rank = B.gauss_jordan(n).first;
        if (rank != n) return {};
        for (int i = 0; i < n; i++) {
            B[i].erase(B[i].begin(), B[i].begin() + n);
        }
        return B;
    }

    std::vector<std::vector<T>> system_of_linear_equations(const std::vector<T>& b) const {
        assert(height() == int(b.size()));
        int n = height(), m = width();
        Matrix B(*this);
        for (int i = 0; i < n; i++) B[i].emplace_back(b[i]);
        int rank = B.gauss_jordan(m).first;
        for (int i = rank; i < n; i++) {
            if (B[i][m] != T(0)) {
                return {};
            }
        }
        std::vector<std::vector<T>> res(1, std::vector<T>(m, 0));
        std::vector<int> pivot(m, -1);
        for (int i = 0, j = 0; i < rank; i++) {
            while (B[i][j] == T(0)) j++;
            res[0][j] = B[i][m];
            pivot[j] = i;
        }
        for (int j = 0; j < m; j++) {
            if (pivot[j] != -1) continue;
            std::vector<T> x(m);
            x[j] = 1;
            for (int k = 0; k < j; k++) {
                if (pivot[k] != -1) {
                    x[k] = -B[pivot[k]][j];
                }
            }
            res.emplace_back(x);
        }
        return res;
    }

    friend std::ostream& operator<<(std::ostream& os, const Matrix& p) {
        int n = p.height(), m = p.width();
        os << "[(" << n << " * " << m << " Matrix)";
        os << "\n[columun sums: ";
        for (int j = 0; j < m; j++) {
            T sum = 0;
            for (int i = 0; i < n; i++) sum += p[i][j];
            os << sum << (j + 1 < m ? "," : "");
        }
        os << "]";
        for (int i = 0; i < n; i++) {
            os << "\n[";
            for (int j = 0; j < m; j++) os << p[i][j] << (j + 1 < m ? "," : "");
            os << "]";
        }
        os << "]\n";
        return os;
    }

  private:
    std::pair<int, T> gauss_jordan(int pivot_end = -1) {
        if (empty()) return {0, T(1)};
        if (pivot_end == -1) pivot_end = width();
        int rank = 0;
        T det = 1;
        for (int j = 0; j < pivot_end; j++) {
            int pivot = -1;
            for (int i = rank; i < height(); i++) {
                if ((*this)[i][j] != T(0)) {
                    pivot = i;
                    break;
                }
            }
            if (pivot == -1) {
                det = 0;
                continue;
            }
            if (pivot != rank) {
                det = -det;
                std::swap((*this)[pivot], (*this)[rank]);
            }
            det *= A[rank][j];
            if (A[rank][j] != T(1)) {
                T coef = T(1) / (*this)[rank][j];
                for (int k = j; k < width(); k++) (*this)[rank][k] *= coef;
            }
            for (int i = 0; i < height(); i++) {
                if (i == rank) continue;
                T coef = (*this)[i][j];
                if (coef == T(0)) continue;
                for (int k = j; k < width(); k++) (*this)[i][k] -= (*this)[rank][k] * coef;
            }
            rank++;
        }
        return {rank, det};
    }
};

struct RandomNumberGenerator {
    std::mt19937 mt;

    RandomNumberGenerator() : mt(std::chrono::steady_clock::now().time_since_epoch().count()) {}

    uint32_t operator()(uint32_t a, uint32_t b) {
        std::uniform_int_distribution<uint32_t> dist(a, b - 1);
        return dist(mt);
    }

    uint32_t operator()(uint32_t b) { return (*this)(0, b); }

    template <typename T> void shuffle(std::vector<T>& v) {
        for (int i = 0; i < int(v.size()); i++) std::swap(v[i], v[(*this)(0, i + 1)]);
    }
};

struct RandomNumberGenerator64 {
    std::mt19937_64 mt;

    RandomNumberGenerator64() : mt(std::chrono::steady_clock::now().time_since_epoch().count()) {}

    uint64_t operator()(uint64_t a, uint64_t b) {
        std::uniform_int_distribution<uint64_t> dist(a, b - 1);
        return dist(mt);
    }

    uint64_t operator()(uint64_t b) { return (*this)(0, b); }

    template <typename T> void shuffle(std::vector<T>& v) {
        for (int i = 0; i < int(v.size()); i++) std::swap(v[i], v[(*this)(0, i + 1)]);
    }
};

using ll = long long;

using namespace std;

constexpr int MOD = 754974721, Root = atcoder::internal::primitive_root_constexpr(MOD), MAX = 5;
using mint = atcoder::static_modint<MOD>;

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);
    cout << fixed << setprecision(15);
    int n;
    cin >> n;
    vector a(n, vector<int>(n));
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            cin >> a[i][j];
        }
    }

    RandomNumberGenerator rng;
    mint root = mint(Root).pow((MOD - 1) / 5);
    vector z(n, vector<mint>(n));
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            z[i][j] = rng(MOD);
        }
    }
    vector<mint> f(MAX);
    for (int k = 0; k < MAX; k++) {
        Matrix<mint> m(n);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                m[i][j] = z[i][j] * root.pow(k * a[i][j]);
            }
        }
        f[k] = m.det();
    }

    for (int i = 0; i < MAX; i++) {
        mint sum = 0;
        for (int j = 0; j < MAX; j++) sum += f[j] * root.pow((MAX - i) * j % MAX);
        cout << (sum != 0 ? "Y" : "N");
    }
    cout << "\n";
    return 0;
}