#include <bits/stdc++.h>
#pragma GCC diagnostic ignored "-Wsign-compare"
#pragma GCC diagnostic ignored "-Wsign-conversion"
#define NDEBUG
#define SHOW(...) static_cast<void>(0)
//!===========================================================!//
//!  dP     dP                          dP                    !//
//!  88     88                          88                    !//
//!  88aaaaa88a .d8888b. .d8888b. .d888b88 .d8888b. 88d888b.  !//
//!  88     88  88ooood8 88'  '88 88'  '88 88ooood8 88'  '88  !//
//!  88     88  88.  ... 88.  .88 88.  .88 88.  ... 88        !//
//!  dP     dP  '88888P' '88888P8 '88888P8 '88888P' dP        !//
//!===========================================================!//
using ld = long double;
using uint = unsigned int;
using ll = long long;
using ull = unsigned long long;
constexpr unsigned int MOD = 1000000007;
template <typename T>
constexpr T INF = std::numeric_limits<T>::max() / 4;
template <typename F>
constexpr F PI = static_cast<F>(3.1415926535897932385);
std::mt19937 mt{std::random_device{}()};
template <typename T>
bool chmin(T& a, const T& b) { return a = std::min(a, b), a == b; }
template <typename T>
bool chmax(T& a, const T& b) { return a = std::max(a, b), a == b; }
template <typename T>
std::vector<T> Vec(const std::size_t n, T v) { return std::vector<T>(n, v); }
template <class... Args>
auto Vec(const std::size_t n, Args... args) { return std::vector<decltype(Vec(args...))>(n, Vec(args...)); }
template <typename T>
constexpr T popCount(const T u)
{
#ifdef __has_builtin
    return u == 0 ? T(0) : (T)__builtin_popcountll(u);
#else
    unsigned long long v = static_cast<unsigned long long>(u);
    return v = (v & 0x5555555555555555ULL) + (v >> 1 & 0x5555555555555555ULL), v = (v & 0x3333333333333333ULL) + (v >> 2 & 0x3333333333333333ULL), v = (v + (v >> 4)) & 0x0F0F0F0F0F0F0F0FULL, static_cast<T>(v * 0x0101010101010101ULL >> 56 & 0x7f);
#endif
}
template <typename T>
constexpr T log2p1(const T u)
{
#ifdef __has_builtin
    return u == 0 ? T(0) : T(64 - __builtin_clzll(u));
#else
    unsigned long long v = static_cast<unsigned long long>(u);
    return v = static_cast<unsigned long long>(v), v |= (v >> 1), v |= (v >> 2), v |= (v >> 4), v |= (v >> 8), v |= (v >> 16), v |= (v >> 32), popCount(v);
#endif
}
template <typename T>
constexpr T clog(const T v) { return v == 0 ? T(0) : log2p1(v - 1); }
template <typename T>
constexpr T msbp1(const T v) { return log2p1(v); }
template <typename T>
constexpr T lsbp1(const T v)
{
#ifdef __has_builtin
    return __builtin_ffsll(v);
#else
    return v == 0 ? T(0) : popCount((v & (-v)) - T(1)) + T(1);
#endif
}
template <typename T>
constexpr bool ispow2(const T v) { return popCount(v) == 1; }
template <typename T>
constexpr T ceil2(const T v) { return v == 0 ? T(1) : T(1) << log2p1(v - 1); }
template <typename T>
constexpr T floor2(const T v) { return v == 0 ? T(0) : T(1) << (log2p1(v) - 1); }
//!===============================================================!//
//!   88888888b            dP       .88888.   a88888b. 888888ba   !//
//!   88                   88      d8'   '88 d8'   '88 88    '8b  !//
//!  a88aaaa    dP.  .dP d8888P    88        88        88     88  !//
//!   88         '8bd8'    88      88   YP88 88        88     88  !//
//!   88         .d88b.    88      Y8.   .88 Y8.   .88 88    .8P  !//
//!   88888888P dP'  'dP   dP       '88888'   Y88888P' 8888888P   !//
//!===============================================================!//
template <typename T>
constexpr std::pair<T, T> extgcd(const T a, const T b)
{
    if (b == 0) { return std::pair<T, T>{1, 0}; }
    const auto p = extgcd(b, a % b);
    return {p.second, p.first - p.second * (a / b)};
}
template <typename T>
constexpr T inverse(const T a, const T mod) { return (mod + extgcd((mod + a % mod) % mod, mod).first % mod) % mod; }
//!========================================================!//
//!  8888ba.88ba                 dP    dP            dP    !//
//!  88  '8b  '8b                88    88            88    !//
//!  88   88   88 .d8888b. .d888b88    88 88d888b. d8888P  !//
//!  88   88   88 88'  '88 88'  '88    88 88'  '88   88    !//
//!  88   88   88 88.  .88 88.  .88    88 88    88   88    !//
//!  dP   dP   dP '88888P' '88888P8    dP dP    dP   dP    !//
//!========================================================!//
template <uint mod>
class ModInt
{
private:
    uint v;
    static uint norm(const uint& x) { return x < mod ? x : x - mod; }
    static ModInt make(const uint& x)
    {
        ModInt m;
        return m.v = x, m;
    }
    static ModInt power(ModInt x, ll n)
    {
        ModInt ans = 1;
        for (; n; n >>= 1, x *= x) {
            if (n & 1) { ans *= x; }
        }
        return ans;
    }
    static ModInt inv(const ModInt& x) { return ModInt{inverse((ll)x.v, (ll)mod)}; }

public:
    ModInt() : v{0} {}
    ModInt(const ll val) : v{norm(uint(val % (ll)mod + (ll)mod))} {}
    ModInt(const ModInt<mod>& n) : v{n()} {}
    explicit operator bool() const { return v != 0; }
    ModInt<mod>& operator=(const ModInt<mod>& n) { return v = n(), (*this); }
    ModInt<mod>& operator=(const ll val) { return v = norm(uint(val % (ll)mod + (ll)mod)), (*this); }
    ModInt<mod> operator+() const { return *this; }
    ModInt<mod> operator-() const { return make(norm(mod - v)); }
    ModInt<mod> operator+(const ModInt<mod>& val) const { return make(norm(v + val())); }
    ModInt<mod> operator-(const ModInt<mod>& val) const { return make(norm(v + mod - val())); }
    ModInt<mod> operator*(const ModInt<mod>& val) const { return make((uint)((ll)v * val() % (ll)mod)); }
    ModInt<mod> operator/(const ModInt<mod>& val) const { return *this * inv(val()); }
    ModInt<mod>& operator+=(const ModInt<mod>& val) { return *this = *this + val; }
    ModInt<mod>& operator-=(const ModInt<mod>& val) { return *this = *this - val; }
    ModInt<mod>& operator*=(const ModInt<mod>& val) { return *this = *this * val; }
    ModInt<mod>& operator/=(const ModInt<mod>& val) { return *this = *this / val; }
    ModInt<mod> operator+(const ll val) const { return ModInt{v + val}; }
    ModInt<mod> operator-(const ll val) const { return ModInt{v - val}; }
    ModInt<mod> operator*(const ll val) const { return ModInt{(ll)v * (val % mod)}; }
    ModInt<mod> operator/(const ll val) const { return ModInt{(ll)v * inv(val)}; }
    template <typename I>
    ModInt<mod> operator^(const I n) const { return power(v, n); }
    ModInt<mod>& operator+=(const ll val) { return *this = *this + val; }
    ModInt<mod>& operator-=(const ll val) { return *this = *this - val; }
    ModInt<mod>& operator*=(const ll val) { return *this = *this * val; }
    ModInt<mod>& operator/=(const ll val) { return *this = *this / val; }
    template <typename I>
    ModInt<mod>& operator^=(const I n) { return (*this) = ((*this) ^ n); }
    bool operator==(const ModInt<mod>& val) const { return v == val.v; }
    bool operator!=(const ModInt<mod>& val) const { return not(*this == val); }
    bool operator==(const ll val) const { return v == norm(uint((ll)mod + val % (ll)mod)); }
    bool operator!=(const ll val) const { return not(*this == val); }
    uint operator()() const { return v; }
};
template <uint mod>
inline ModInt<mod> operator+(const ll val, const ModInt<mod>& n) { return n + val; }
template <uint mod>
inline ModInt<mod> operator-(const ll val, const ModInt<mod>& n) { return ModInt<mod>{val - (ll)n()}; }
template <uint mod>
inline ModInt<mod> operator*(const ll val, const ModInt<mod>& n) { return n * val; }
template <uint mod>
inline ModInt<mod> operator/(const ll val, const ModInt<mod>& n) { return ModInt<mod>(val) / n; }
template <uint mod>
inline bool operator==(const ll val, const ModInt<mod>& n) { return n == val; }
template <uint mod>
inline bool operator!=(const ll val, const ModInt<mod>& n) { return not(val == n); }
template <uint mod>
inline std::istream& operator>>(std::istream& is, ModInt<mod>& n)
{
    uint v;
    return is >> v, n = v, is;
}
template <uint mod>
std::ostream& operator<<(std::ostream& os, const ModInt<mod>& n) { return (os << n()); }
//!============================================================================!//
//!  8888ba.88ba                 dP     a88888b.                     dP        !//
//!  88  '8b  '8b                88    d8'   '88                     88        !//
//!  88   88   88 .d8888b. .d888b88    88        .d8888b. 88d8b.d8b. 88d888b.  !//
//!  88   88   88 88'  '88 88'  '88    88        88'  '88 88''88''88 88'  '88  !//
//!  88   88   88 88.  .88 88.  .88    Y8.   .88 88.  .88 88  88  88 88.  .88  !//
//!  dP   dP   dP '88888P' '88888P8     Y88888P' '88888P' dP  dP  dP 88Y8888'  !//
//!============================================================================!//
template <uint mod>
class ModComb
{
public:
    ModComb(const std::size_t N) : f(N + 1, ModInt<mod>(1)), in(N + 1, ModInt<mod>(1)), invf(N + 1, ModInt<mod>(1))
    {
        for (uint i = 2; i <= N; i++) { f[i] = f[i - 1] * i, in[i] = (mod - (mod / i)) * in[mod % i], invf[i] = invf[i - 1] * in[i]; }
    }
    ModInt<mod> fact(const std::size_t N) const { return f[N]; }
    ModInt<mod> inv(const std::size_t N) const { return in[N]; }
    ModInt<mod> invFact(const std::size_t N) const { return invf[N]; }
    ModInt<mod> perm(const std::size_t N, const std::size_t K) const { return N > f.size() or K > N ? ModInt<mod>(0) : f[N] * invf[N - K]; }
    ModInt<mod> comb(const std::size_t N, const std::size_t K) const { return N > f.size() or K > N ? ModInt<mod>(0) : f[N] * invf[K] * invf[N - K]; }

private:
    std::vector<ModInt<mod>> f, in, invf;
};
template <typename F>
struct Complex
{
    F x, y;
    Complex() : x(0), y(0) {}
    Complex(const F& s) : x(std::cos(s)), y(std::sin(s)) {}
    Complex(const F x, const F y) : x(x), y(y) {}
    Complex operator-() const { return Complex(-x, -y); }
    Complex operator+(const Complex& c) const { return Complex{x + c.x, y + c.y}; }
    Complex operator-(const Complex& c) const { return Complex{x - c.x, y - c.y}; }
    Complex operator*(const Complex& c) const { return Complex{x * c.x - y * c.y, x * c.y + y * c.x}; }
    Complex operator*(const F& r) const { return Complex{x * r, y * r}; }
    Complex& operator+=(const Complex& c) { return this->x += c.x, this->y += c.y, *this; }
    Complex& operator-=(const Complex& c) { return this->x -= c.x, this->y -= c.y, *this; }
    Complex& operator*=(const Complex& c) { return *this = *this * c; }
    Complex& operator*=(const F& r) { return this->x *= r, this->y *= r, *this; }
    Complex conj() const { return Complex{x, -y}; }
};
//!==================================!//
//!   88888888b  88888888b d888888P  !//
//!   88         88           88     !//
//!  a88aaaa    a88aaaa       88     !//
//!   88         88           88     !//
//!   88         88           88     !//
//!   dP         dP           dP     !//
//!==================================!//
template <typename F = double>
class FFT
{
private:
    static constexpr std::size_t L = 30;

public:
    FFT() = delete;
    static void fft(std::vector<Complex<F>>& a, const std::size_t lg, const bool rev)
    {
        static std::vector<Complex<F>> root[L];
        const std::size_t N = a.size();
        assert((1UL << lg) == N);
        if (root[lg].empty()) {
            root[lg].reserve(N), root[lg].resize(N);
            for (std::size_t i = 0; i < N; i++) { root[lg][i] = Complex<F>(PI<F> * F(2 * i) / F(N)); }
        }
        std::vector<Complex<F>> tmp(N);
        for (std::size_t w = (N >> 1); w > 0; w >>= 1) {
            for (std::size_t y = 0; y < (N >> 1); y += w) {
                const Complex<F> r = rev ? root[lg][y].conj() : root[lg][y];
                for (std::size_t x = 0; x < w; x++) {
                    const auto u = a[y << 1 | x], v = a[y << 1 | x | w] * r;
                    tmp[y | x] = u + v, tmp[y | x | (N >> 1)] = u - v;
                }
            }
            std::swap(tmp, a);
        }
    }
    template <typename T = ll, typename I = int>
    static std::vector<T> simpleConvolute(const std::vector<I>& a, const std::vector<I>& b)
    {
        const std::size_t need = a.size() + b.size() - 1, lg = clog(need), N = 1UL << lg;
        std::vector<Complex<F>> A(N), B(N);
        for (std::size_t i = 0; i < a.size(); i++) { A[i] = Complex<F>{F(a[i]), 0}; }
        for (std::size_t i = 0; i < b.size(); i++) { B[i] = Complex<F>{F(b[i]), 0}; }
        fft(A, lg, false), fft(B, lg, false);
        for (std::size_t i = 0; i < N; i++) { A[i] *= B[i] * ((F)1 / (F)N); }
        fft(A, lg, true);
        std::vector<T> ans(need);
        for (std::size_t i = 0; i < need; i++) { ans[i] = T(std::round(A[i].x)); }
        return ans;
    }
    template <typename T = ll, std::size_t K = 2, typename I = int>
    static std::vector<T> convolute(const std::vector<I>& a, const std::vector<I>& b)
    {
        constexpr std::size_t V = 30;
        constexpr std::size_t S = (V + K - 1) / K;
        const std::size_t need = a.size() + b.size() - 1, lg = clog(need), N = 1UL << lg;
        std::vector<Complex<F>> A[K], B[K], tmp(N);
        for (std::size_t i = 0; i < K; i++) {
            A[i].reserve(N), A[i].resize(N), B[i].reserve(N), B[i].resize(N);
            std::fill(tmp.begin() + std::min(a.size(), b.size()), tmp.end(), Complex<F>{});
            for (std::size_t j = 0; j < a.size(); j++) { tmp[j].x = F((a[j] >> (S * i)) & ((1 << S) - 1)); }
            for (std::size_t j = 0; j < b.size(); j++) { tmp[j].y = F((b[j] >> (S * i)) & ((1 << S) - 1)); }
            fft(tmp, lg, false);
            for (std::size_t j = 0; j < N; j++) { tmp[j] *= F(0.5); }
            for (std::size_t j = 0; j < N; j++) {
                const std::size_t k = j == 0 ? 0UL : N - j;
                A[i][j] = Complex<F>{tmp[j].x + tmp[k].x, tmp[j].y - tmp[k].y}, B[i][j] = Complex<F>{tmp[j].y + tmp[k].y, -tmp[j].x + tmp[k].x};
            }
        }
        std::vector<Complex<F>> Z[K];
        for (std::size_t i = 0; i < K; i++) { Z[i].reserve(N), Z[i].resize(N); }
        for (std::size_t a = 0; a < K; a++) {
            for (std::size_t b = 0; b < K; b++) {
                for (std::size_t i = 0; i < N; i++) {
                    if (a + b < K) {
                        Z[a + b][i] += A[a][i] * B[b][i];
                    } else {
                        Z[a + b - K][i] += A[a][i] * B[b][i] * Complex<F>(0, 1);
                    }
                }
            }
        }
        for (std::size_t i = 0; i < K; i++) { fft(Z[i], lg, true); }
        std::vector<T> ans(need);
        T base = 1;
        for (std::size_t k = 0; k < 2 * K - 1; k++, base *= (1LL << S)) {
            for (std::size_t i = 0; i < need; i++) {
                if (k < K) {
                    ans[i] += base * T(std::round(Z[k][i].x / F(N)));
                } else {
                    ans[i] += base * T(std::round(Z[k - K][i].y / F(N)));
                }
            }
        }
        return ans;
    }
    template <uint mod, std::size_t K = 2>
    static std::vector<ModInt<mod>> convolute(const std::vector<ModInt<mod>>& a, const std::vector<ModInt<mod>>& b)
    {
        constexpr std::size_t V = 30;
        constexpr std::size_t S = (V + K - 1) / K;
        const std::size_t need = a.size() + b.size() - 1, lg = clog(need), N = 1UL << lg;
        std::vector<Complex<F>> A[K], B[K], tmp(N);
        for (std::size_t i = 0; i < K; i++) {
            A[i].reserve(N), A[i].resize(N), B[i].reserve(N), B[i].resize(N);
            std::fill(tmp.begin() + std::min(a.size(), b.size()), tmp.end(), Complex<F>{});
            for (std::size_t j = 0; j < a.size(); j++) { tmp[j].x = F((a[j]() >> (S * i)) & ((1 << S) - 1)); }
            for (std::size_t j = 0; j < b.size(); j++) { tmp[j].y = F((b[j]() >> (S * i)) & ((1 << S) - 1)); }
            fft(tmp, lg, false);
            for (std::size_t j = 0; j < N; j++) { tmp[j] *= F(0.5); }
            for (std::size_t j = 0; j < N; j++) {
                const std::size_t k = j == 0 ? 0UL : N - j;
                A[i][j] = Complex<F>{tmp[j].x + tmp[k].x, tmp[j].y - tmp[k].y}, B[i][j] = Complex<F>{tmp[j].y + tmp[k].y, -tmp[j].x + tmp[k].x};
            }
        }
        std::vector<Complex<F>> Z[K];
        for (std::size_t i = 0; i < K; i++) { Z[i].reserve(N), Z[i].resize(N); }
        for (std::size_t a = 0; a < K; a++) {
            for (std::size_t b = 0; b < K; b++) {
                for (std::size_t i = 0; i < N; i++) {
                    if (a + b < K) {
                        Z[a + b][i] += A[a][i] * B[b][i];
                    } else {
                        Z[a + b - K][i] += A[a][i] * B[b][i] * Complex<F>(0, 1);
                    }
                }
            }
        }
        for (std::size_t i = 0; i < K; i++) { fft(Z[i], lg, true); }
        std::vector<ModInt<mod>> ans(need);
        ModInt<mod> base = 1;
        for (std::size_t k = 0; k < 2 * K - 1; k++, base *= (1LL << S)) {
            for (std::size_t i = 0; i < need; i++) {
                if (k < K) {
                    ans[i] += int((base * ll(std::round(Z[k][i].x / F(N))))());
                } else {
                    ans[i] += int((base * ll(std::round(Z[k - K][i].y / F(N))))());
                }
            }
        }
        return ans;
    }
};
//!===================================!//
//!   888888ba           dP           !//
//!   88    '8b          88           !//
//!  a88aaaa8P' .d8888b. 88 dP    dP  !//
//!   88        88'  '88 88 88    88  !//
//!   88        88.  .88 88 88.  .88  !//
//!   dP        '88888P' dP '8888P88  !//
//!                              .88  !//
//!                          d8888P   !//
//!===================================!//
template <uint mod, uint K = 2>
class Poly
{
public:
    using mint = ModInt<mod>;
    Poly<mod> rev(const std::size_t l) const
    {
        std::vector<mint> ans = v;
        ans.resize(l);
        std::reverse(ans.begin(), ans.end());
        return Poly(ans);
    }
    Poly() : v(0) {}
    Poly(const mint& r) : v{r} { shrink(); }
    Poly(const std::vector<mint>& v) : v{v} { shrink(); }
    Poly(const std::initializer_list<mint>&& list) : v{list} { shrink(); }
    std::vector<mint> operator()() const { return v; }
    mint& operator[](const std::size_t i) { return v[i]; }
    const mint& operator[](const std::size_t i) const { return v[i]; }
    mint at(const std::size_t i) const { return i < size() ? v[i] : mint(0); }
    Poly<mod> operator-() const
    {
        std::vector<mint> ans = v;
        for (auto& e : ans) { e = -e; }
        return Poly<mod>(ans);
    }
    Poly<mod> operator+(const Poly<mod>& p) const
    {
        const std::size_t sz = std::max(p.size(), size());
        std::vector<mint> ans(sz);
        for (std::size_t i = 0; i < sz; i++) { ans[i] = at(i) + p.at(i); }
        return Poly<mod>(ans);
    }
    Poly<mod> operator-(const Poly<mod>& p) const
    {
        const std::size_t sz = std::max(p.size(), size());
        std::vector<mint> ans(sz);
        for (std::size_t i = 0; i < sz; i++) { ans[i] = at(i) - p.at(i); }
        return Poly<mod>(ans);
    }
    Poly<mod> operator*(const Poly<mod>& p) const { return p.size() == 0 or size() == 0 ? Poly() : Poly(multiply(v, p())); }
    Poly<mod> operator*(const mint& r) const
    {
        std::vector<mint> ans = v;
        for (auto& e : ans) { e *= r; }
        return Poly(ans);
    }
    Poly<mod> operator/(const mint& r) const
    {
        std::vector<mint> ans = v;
        for (auto& e : ans) { e /= r; }
        return Poly(ans);
    }
    Poly<mod> operator<<(const std::size_t s) const
    {
        const std::size_t N = size();
        if (N <= s) { return Poly(); }
        std::vector<mint> ans(N - s);
        for (std::size_t i = 0; i < N - s; i++) { ans[i] = v[i + s]; }
        return Poly(ans);
    }
    Poly<mod> operator>>(const std::size_t s) const
    {
        const std::size_t N = size();
        if (N == 0) { return Poly(); }
        std::vector<mint> ans(N + s);
        for (std::size_t i = 0; i < N; i++) { ans[i + s] = v[i]; }
        return Poly(ans);
    }
    Poly<mod> operator/(const Poly<mod>& p) const { return div(p); }
    Poly<mod> operator%(const Poly<mod>& p) const { return rem(p); }
    Poly<mod>& operator+=(const Poly<mod>& p) { return *this = (*this + p); }
    Poly<mod>& operator-=(const Poly<mod>& p) { return *this = (*this - p); }
    Poly<mod>& operator*=(const Poly<mod>& p) { return *this = Poly(multiply(v, p())); }
    Poly<mod>& operator*=(const mint& r)
    {
        for (auto& e : v) { e *= r; }
    }
    Poly<mod>& operator/=(const mint& r)
    {
        for (auto& e : v) { e /= r; }
    }
    Poly<mod>& operator>>=(const std::size_t s) { return *this = (*this >> s); }
    Poly<mod>& operator<<=(const std::size_t s) { return *this = (*this << s); }
    Poly<mod>& operator/=(const Poly<mod>& p) { return *this = div(p); }
    Poly<mod>& operator%=(const Poly<mod>& p) { return *this = rem(p); }
    Poly<mod> rem(const std::size_t k) const { return size() <= k ? *this : Poly(std::vector<mint>(v.begin(), v.begin() + k)); }
    Poly<mod> rem(const Poly<mod>& q) const { return *this - div(q) * q; }
    Poly<mod> rem(const Poly<mod>& q, const Poly<mod>& iq, const std::size_t B) { return *this - q * ((*this * iq) >> (B - 1)); }
    Poly<mod> inv(const std::size_t k) const
    {
        Poly<mod> q(std::vector<mint>(1, mint(1) / v[0]));
        const auto T = Poly(std::vector<mint>{2});
        for (std::size_t i = 1, j = 0; j < k; j++, i *= 2) { q = (q * (T - rem(2 * i) * q)).rem(2 * i); }
        return q;
    }
    template <typename I>
    Poly<mod> power(const I k) const
    {
        using mint = ModInt<mod>;
        const std::size_t B = size() * 2 - 1;
        const auto q = pseudoInv(B);
        Poly<mod> ans(std::vector<mint>{1});
        const std::size_t D = log2p1<std::size_t>(k);
        for (std::size_t i = 0; i < D; i++) {
            if (k & (1LL << (D - i - 1))) { ans = (ans << 1).rem(*this, q, B); }
            if (D - i - 1) { ans = (ans * ans).rem(*this, q, B); }
        }
        return ans;
    }
    std::size_t size() const { return v.size(); }
    friend std::ostream& operator<<(std::ostream& os, const Poly& p)
    {
        if (p.size() == 0) { return os << "0"; }
        for (std::size_t i = 0; i < p.size(); i++) { os << (i != 0 ? "+" : "") << p[i] << (i != 0 ? i == 1 ? "X" : "X^" + std::to_string(i) : ""); }
        return os;
    }

private:
    Poly<mod> div(const Poly<mod>& q) const
    {
        assert(q.size() > 0);
        if (size() < q.size()) { return Poly(); }
        const std::size_t N = size();
        const auto iq = q.pseudoInv(N);
        return (*this * iq) >> (N - 1);
    }
    void shrink()
    {
        for (; not v.empty() and v.back() == 0; v.pop_back()) {}
    }
    static std::vector<mint> multiply(const std::vector<mint>& a, const std::vector<mint>& b) { return FFT<double>::convolute<mod, K>(a, b); }
    Poly<mod> pseudoInv(const std::size_t B) const
    {
        const std::size_t N = size();
        return rev(N).inv(B + 2 > N ? clog(B - N + 2) : 0).rev(B + 1 - N);
    }

    std::vector<mint> v;
};
//!============================================!//
//!    8888ba.88ba             oo              !//
//!    88  '8b  '8b                            !//
//!    88   88   88  .d8888b.  dP  88d888b.    !//
//!    88   88   88  88'  '88  88  88'  '88    !//
//!    88   88   88  88.  .88  88  88    88    !//
//!    dP   dP   dP  '88888P8  dP  dP    dP    !//
//!============================================!//
using mint = ModInt<MOD>;
int main()
{
    int N;
    uint B;
    std::cin >> N >> B;
    assert(1 <= N and N <= 200000);
    assert(1 <= B and B < MOD);
    std::vector<int> c(N + 1, 0);
    for (int i = 0; i < N; i++) {
        int S;
        std::cin >> S;
        assert(0 <= S and S < N);
        c[S]++;
    }
    std::vector<int> C = c;
    for (int i = N - 1; i >= 0; i--) { C[i] += C[i + 1]; }
    ModComb<MOD> modcomb(N);
    std::queue<Poly<MOD>> Q;
    for (int i = 0; i < N; i++) {
        if (c[N - i - 1] == 0) { continue; }
        const mint alpha = modcomb.fact(C[N - i - 1]) * modcomb.invFact(C[N - i]) * c[N - i - 1] * modcomb.inv(C[N - i - 1]);
        const mint beta = modcomb.fact(C[N - i - 1]) * modcomb.invFact(C[N - i]) * C[N - i] * modcomb.inv(C[N - i - 1]);
        Q.push({beta, alpha});
    }
    while (Q.size() > 1) {
        const auto f1 = Q.front();
        Q.pop();
        const auto f2 = Q.front();
        Q.pop();
        Q.push(f1 * f2);
    }
    const auto P = Q.front();
    mint ans = 0;
    mint base = 1;
    for (int i = 0; i <= N; i++, base *= B) { ans += i * base * P.at(i); }
    std::cout << ans << std::endl;
    return 0;
}