#include <bits/stdc++.h>
#pragma GCC diagnostic ignored "-Wsign-compare"
#pragma GCC diagnostic ignored "-Wsign-conversion"
//!===========================================================!//
//!  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 ll = long long;
using ull = unsigned long long;
std::mt19937 mt{std::random_device{}()};
template <typename T>
constexpr T INF = std::numeric_limits<T>::max() / 4;
template <typename T>
constexpr T MOD = static_cast<T>(1000000007);
template <typename F>
constexpr F PI() { return 3.1415926535897932385; }
#define SHOW(...) (std::cerr << "(" << #__VA_ARGS__ << ") = ("), HogeHogeSansuu(__VA_ARGS__), std::cerr << ")" << std::endl;
struct has_debugPrint_impl
{
    template <class T>
    static auto check(T&& x) -> decltype(x.debugPrint(), std::true_type{});
    template <class T>
    static auto check(...) -> std::false_type;
};
template <class T>
class has_debugPrint : public decltype(has_debugPrint_impl::check<T>(std::declval<T>()))
{
};
template <bool>
struct HogeHogeDump
{
    template <typename T>
    static void dump(const T& x) { x.debugPrint(); }
};
template <>
struct HogeHogeDump<false>
{
    template <typename T>
    static void dump(const T& x) { std::cerr << x; }
};
void HogeHogeSansuu() { ; }
template <typename T>
void HogeHogeSansuu(const T& x) { HogeHogeDump<has_debugPrint<T>::value>::dump(x); }
template <typename T, typename... Args>
void HogeHogeSansuu(const T& x, Args... args) { HogeHogeDump<has_debugPrint<T>::value>::dump(x), std::cerr << ",", HogeHogeSansuu(args...); }
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, typename F>
void For(const T s, const T t, const F f)
{
    for (T i = s; i != t; i += T(s < t ? 1 : -1)) { f(i); }
}
template <typename T, typename F>
void Rep(const T N, const F f) { For<T, F>(0, N, f); }
template <typename T, typename F>
void RRep(const T N, const F f) { For<T, F>(N - 1, -1, f); }
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(T v) { 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); }
template <typename T>
constexpr T log2p1(T v) { return v |= (v >> 1), v |= (v >> 2), v |= (v >> 4), v |= (v >> 8), v |= (v >> 16), v |= (v >> 32), PopCount(v); }
template <typename T>
constexpr bool ispow2(const T v) { return (v << 1) == (T(1) << (log2p1(v))); }
template <typename T>
constexpr T ceil2(const T v) { return ispow2(v) ? v : T(1) << log2p1(v); }
template <typename T>
constexpr T floor2(const T v) { return v == 0 ? T(0) : ispow2(v) ? v : T(1) << (log2p1(v) - 1); }
template <typename T>
struct Accum
{
    template <typename InIt>
    Accum(const InIt first, const InIt last) : accum(std::size_t(std::distance(first, last))) { std::partial_sum(first, last, accum.begin()); }
    T sum(const std::size_t i) const { return i == 0 ? T(0) : accum[i - 1]; }
    T sum(const std::size_t l, const std::size_t r) const { return sum(r) - sum(l); }
    std::vector<T> accum;
};
template <typename T>
struct Accum2D
{
    Accum2D(const std::vector<std::vector<T>>& t) : accum{t}
    {
        for (std::size_t i = 0; i < accum.size(); i++) {
            for (std::size_t j = 1; j < accum[i].size(); j++) { accum[i][j] += accum[i][j - 1]; }
        }
        for (std::size_t i = 1; i < accum.size(); i++) {
            for (std::size_t j = 0; j < accum[i].size(); j++) { accum[i][j] += accum[i - 1][j]; }
        }
    }
    T sum(const std::size_t y, const std::size_t x) const { return y == 0 or x == 0 ? T(0) : accum[y - 1][x - 1]; }
    T sum(const std::size_t ymin, const std::size_t ysup, const std::size_t xmin, const std::size_t xsup) const { return sum(ysup, xsup) - sum(ymin, xmin); }
    std::vector<std::vector<T>> accum;
};
template <typename T>
struct Zip
{
    template <typename InIt>
    Zip(const InIt first, const InIt last) : unzip(std::size_t(std::distance(first, last)))
    {
        std::copy(first, last, unzip), std::sort(unzip.begin(), unzip.end()), unzip.erase(std::unique(unzip.begin(), unzip.end()), unzip.end());
        for (std::size_t i = 0; i < unzip.size(); i++) { zip[unzip[i]] = i; }
    }
    std::vector<T> unzip;
    std::map<T, std::size_t> zip;
};
template <typename T, std::size_t N>
std::ostream& operator<<(std::ostream& os, const std::array<T, N>& v)
{
    os << "[";
    for (const auto& e : v) { os << e << ","; }
    return (os << "]" << std::endl);
}
template <typename T, typename A>
std::ostream& operator<<(std::ostream& os, const std::deque<T, A>& v)
{
    os << "[";
    for (const auto& e : v) { os << e << ","; }
    return (os << "]" << std::endl);
}
template <typename K, typename T, typename C, typename A>
std::ostream& operator<<(std::ostream& os, const std::multimap<K, T, C, A>& v)
{
    os << "[";
    for (const auto& e : v) { os << "<" << e.first << ": " << e.second << ">,"; }
    return (os << "]" << std::endl);
}
template <typename T, typename C, typename A>
std::ostream& operator<<(std::ostream& os, const std::multiset<T, C, A>& v)
{
    os << "[";
    for (const auto& e : v) { os << e << ","; }
    return (os << "]" << std::endl);
}
template <typename K, typename T, typename C, typename A>
std::ostream& operator<<(std::ostream& os, const std::map<K, T, C, A>& v)
{
    os << "[";
    for (const auto& e : v) { os << "<" << e.first << ": " << e.second << ">,"; }
    return (os << "]" << std::endl);
}
template <typename T1, typename T2>
std::ostream& operator<<(std::ostream& os, const std::pair<T1, T2>& v) { return (os << "<" << v.first << "," << v.second << ">"); }
template <typename T1, typename T2, typename T3>
std::ostream& operator<<(std::ostream& os, const std::priority_queue<T1, T2, T3>& v)
{
    auto q = v;
    os << "[";
    while (not q.empty()) { os << q.top() << ",", q.pop(); }
    return os << "]\n";
}
template <typename T1, typename T2>
std::ostream& operator<<(std::ostream& os, const std::queue<T1>& v)
{
    auto q = v;
    os << "[";
    while (not q.empty()) { os << q.front() << ",", q.pop(); }
    return os << "]\n";
}
template <typename T, typename C, typename A>
std::ostream& operator<<(std::ostream& os, const std::set<T, C, A>& v)
{
    os << "[";
    for (const auto& e : v) { os << e << ","; }
    return (os << "]" << std::endl);
}
template <typename T1, typename T2>
std::ostream& operator<<(std::ostream& os, const std::stack<T1>& v)
{
    auto q = v;
    os << "[";
    while (not q.empty()) { os << q.top() << ",", q.pop(); }
    return os << "]\n";
}
template <typename K, typename T, typename H, typename P, typename A>
std::ostream& operator<<(std::ostream& os, const std::unordered_multimap<K, T, H, P, A>& v)
{
    os << "[";
    for (const auto& e : v) { os << "<" << e.first << ": " << e.second << ">,"; }
    return (os << "]" << std::endl);
}
template <typename T, typename H, typename P, typename A>
std::ostream& operator<<(std::ostream& os, const std::unordered_multiset<T, H, P, A>& v)
{
    os << "[";
    for (const auto& e : v) { os << e << ","; }
    return (os << "]" << std::endl);
}
template <typename K, typename T, typename H, typename P, typename A>
std::ostream& operator<<(std::ostream& os, const std::unordered_map<K, T, H, P, A>& v)
{
    os << "[";
    for (const auto& e : v) { os << "<" << e.first << ": " << e.second << ">,"; }
    return (os << "]" << std::endl);
}
template <typename T, typename H, typename P, typename A>
std::ostream& operator<<(std::ostream& os, const std::unordered_set<T, H, P, A>& v)
{
    os << "[";
    for (const auto& e : v) { os << e << ","; }
    return (os << "]" << std::endl);
}
template <typename T, typename A>
std::ostream& operator<<(std::ostream& os, const std::vector<T, A>& v)
{
    os << "[";
    for (const auto& e : v) { os << e << ","; }
    return (os << "]" << std::endl);
}
//!===============================================================!//
//!   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 <typename T, T mod>
class ModInt
{
private:
    T value;

public:
    ModInt() : value{0} {}
    ModInt(const T val) : value{((val % mod) + mod) % mod} {}
    ModInt(const ModInt<T, mod>& n) : value{n()} {}
    ModInt<T, mod>& operator=(const ModInt<T, mod>& n) { return value = n(), (*this); }
    ModInt<T, mod>& operator=(const T v) { return value = (mod + v % mod) % mod, (*this); }
    ModInt<T, mod> operator+() const { return *this; }
    ModInt<T, mod> operator-() const { return ModInt{mod - value}; }
    ModInt<T, mod> operator+(const ModInt<T, mod>& val) const { return ModInt{value + val()}; }
    ModInt<T, mod> operator-(const ModInt<T, mod>& val) const { return ModInt{value - val() + mod}; }
    ModInt<T, mod> operator*(const ModInt<T, mod>& val) const { return ModInt{value * val()}; }
    ModInt<T, mod> operator/(const ModInt<T, mod>& val) const { return ModInt{value * inverse(val(), mod)}; }
    ModInt<T, mod>& operator+=(const ModInt<T, mod>& val) { return (((value += val()) %= mod) += mod) %= mod, (*this); }
    ModInt<T, mod>& operator-=(const ModInt<T, mod>& val) { return (((value -= val()) %= mod) += mod) %= mod, (*this); }
    ModInt<T, mod>& operator*=(const ModInt<T, mod>& val) { return (((value *= val()) %= mod) += mod) %= mod, (*this); }
    ModInt<T, mod>& operator/=(const ModInt<T, mod>& val) { return (((value *= inverse(val(), mod)) %= mod) += mod) %= mod, (*this); }
    ModInt<T, mod> operator+(const T val) const { return ModInt{value + val}; }
    ModInt<T, mod> operator-(const T val) const { return ModInt{value - val}; }
    ModInt<T, mod> operator*(const T val) const { return ModInt{value * val}; }
    ModInt<T, mod> operator/(const T val) const { return ModInt{value * inverse(val, mod)}; }
    template <typename I>
    ModInt<T, mod> operator^(const I n) const { return n < 0 ? (T(1) / (*this)) ^ (-n) : n == 0 ? ModInt<T, mod>(1) : n % 2 == 1 ? (*this) * ((*this) ^ (n - 1)) : ((*this * *this) ^ (n / 2)); }
    ModInt<T, mod>& operator+=(const T val) { return (((value += val) %= mod) += mod) %= mod, (*this); }
    ModInt<T, mod>& operator-=(const T val) { return (((value -= val) %= mod) += mod) %= mod, (*this); }
    ModInt<T, mod>& operator*=(const T val) { return (((value *= val) %= mod) += mod) %= mod, (*this); }
    ModInt<T, mod>& operator/=(const T val) { return (((value *= inverse(val, mod)) %= mod) += mod) %= mod, (*this); }
    template <typename I>
    ModInt<T, mod>& operator^=(const I n) { return (*this) = ((*this) ^ n); }
    bool operator==(const ModInt<T, mod>& val) const { return value == val.value; }
    bool operator!=(const ModInt<T, mod>& val) const { return not(*this == val); }
    bool operator==(const T val) const { return value == (mod + val % mod) % mod; }
    bool operator!=(const T val) const { return not(*this == val); }
    T operator()() const { return value; }
};
template <typename T, T mod>
inline ModInt<T, mod> operator+(const T val, const ModInt<T, mod>& n) { return ModInt<T, mod>{n() + val}; }
template <typename T, T mod>
inline ModInt<T, mod> operator-(const T val, const ModInt<T, mod>& n) { return ModInt<T, mod>{-n() + val}; }
template <typename T, T mod>
inline ModInt<T, mod> operator*(const T val, const ModInt<T, mod>& n) { return ModInt<T, mod>{n() * val}; }
template <typename T, T mod>
inline ModInt<T, mod> operator/(const T val, const ModInt<T, mod>& n) { return ModInt<T, mod>(val) / n; }
template <typename T, T mod>
inline bool operator==(const T val, const ModInt<T, mod>& n) { return n == val; }
template <typename T, T mod>
inline bool operator!=(const T val, const ModInt<T, mod>& n) { return not(val == n); }
template <typename T, T mod>
inline std::istream& operator>>(std::istream& is, ModInt<T, mod>& n)
{
    T v;
    return is >> v, n = v, is;
}
template <typename T, T mod>
std::ostream& operator<<(std::ostream& os, const ModInt<T, mod>& n) { return (os << n()); }
template <int mod>
using Mint = ModInt<int, mod>;
template <ll mod>
using Mll = ModInt<ll, mod>;
//!=====================================================!//
//!  8888ba.88ba             dP            oo           !//
//!  88  '8b  '8b            88                         !//
//!  88   88   88 .d8888b. d8888P 88d888b. dP dP.  .dP  !//
//!  88   88   88 88'  '88   88   88'  '88 88  '8bd8'   !//
//!  88   88   88 88.  .88   88   88       88  .d88b.   !//
//!  dP   dP   dP '88888P8   dP   dP       dP dP'  'dP  !//
//!=====================================================!//
template <typename T>
struct Matrix
{
    Matrix(const std::size_t R, const std::size_t C) : R{R}, C{C}, table(R, std::vector<T>(C, T(0))) {}
    const std::vector<T>& operator[](const std::size_t r) const { return table[r]; }
    std::vector<T>& operator[](const std::size_t r) { return table[r]; }
    Matrix<T> operator-() const
    {
        Matrix<T> ans(R, C);
        for (std::size_t i = 0; i < R; i++) {
            for (std::size_t j = 0; j < C; j++) { ans[i][j] = -table[i][j]; }
        }
        return ans;
    }
    Matrix<T> operator+(const Matrix<T>& m) const
    {
        assert(R == m.R and C == m.C);
        Matrix<T> ans(R, C);
        for (std::size_t i = 0; i < R; i++) {
            for (std::size_t j = 0; j < C; j++) { ans[i][j] = table[i][j] + m[i][j]; }
        }
        return ans;
    }
    Matrix<T> operator-(const Matrix<T>& m) const
    {
        assert(R == m.R and C == m.C);
        Matrix<T> ans(R, C);
        for (std::size_t i = 0; i < R; i++) {
            for (std::size_t j = 0; j < C; j++) { ans[i][j] = table[i][j] - m[i][j]; }
        }
        return ans;
    }
    Matrix<T> operator*(const Matrix<T>& m) const
    {
        assert(C == m.R);
        Matrix<T> ans(R, m.C);
        for (std::size_t i = 0; i < R; i++) {
            for (std::size_t j = 0; j < m.C; j++) {
                for (std::size_t k = 0; k < C; k++) { ans[i][j] += table[i][k] * m[k][j]; }
            }
        }
        return ans;
    }
    Matrix<T> operator*(const T& t) const
    {
        Matrix<T> ans(R, C);
        for (std::size_t i = 0; i < R; i++) {
            for (std::size_t j = 0; j < C; j++) { ans[i][j] = table[i][j] * t; }
        }
        return ans;
    }
    template <typename C>
    Matrix<T> operator^(const C n) const { return assert(n > 0), n == 1 ? *this : n % 2 == 1 ? (*this) * ((*this) ^ (n - 1)) : (((*this) * (*this)) ^ (n / 2)); }
    Matrix<T>& operator=(const Matrix<T>& m)
    {
        assert(R == m.R and C == m.C);
        for (std::size_t i = 0; i < R; i++) {
            for (std::size_t j = 0; j < C; j++) { table[i][j] = m[i][j]; }
        }
        return *this;
    }
    Matrix<T>& operator+=(const Matrix<T>& m)
    {
        assert(R == m.R and C == m.C);
        for (std::size_t i = 0; i < R; i++) {
            for (std::size_t j = 0; j < C; j++) { table[i][j] += m[i][j]; }
        }
    }
    Matrix<T>& operator-=(const Matrix<T>& m)
    {
        assert(R == m.R and C == m.C);
        for (std::size_t i = 0; i < R; i++) {
            for (std::size_t j = 0; j < C; j++) { table[i][j] -= m[i][j]; }
        }
        return *this;
    }
    Matrix<T>& operator*=(const Matrix<T>& m) { return *this = (*this) * m; }
    Matrix<T>& operator*=(const T& t)
    {
        for (std::size_t i = 0; i < R; i++) {
            for (std::size_t j = 0; j < C; j++) { table[i][j] *= t; }
        }
        return *this;
    }
    template <typename C>
    Matrix<T>& operator^=(const C n) { return (*this) = ((*this) ^ n); }
    const std::size_t R, C;
    std::vector<std::vector<T>> table;
};
template <typename T>
Matrix<T> operator*(const T& t, const Matrix<T>& m) { return m * t; }
template <typename T>
std::ostream& operator<<(std::ostream& os, const Matrix<T>& m)
{
    os << "[\n";
    for (std::size_t i = 0; i < m.R; i++) {
        os << "[";
        for (std::size_t j = 0; j < m.C; j++) { os << m[i][j] << " "; }
        os << "]\n";
    }
    return (os << "]\n");
}
template <typename T>
struct Vector
{
    Vector(const std::size_t R) : R{R}, table(R, T(0)) {}
    const T& operator[](const std::size_t r) const { return table[r]; }
    T& operator[](const std::size_t r) { return table[r]; }
    Vector<T> operator+(const Vector<T>& v) const
    {
        assert(R == v.R);
        Vector<T> ans(R);
        for (std::size_t i = 0; i < R; i++) { ans[i] = table[i] + v[i]; }
        return ans;
    }
    Vector<T> operator-(const Vector<T>& v) const
    {
        assert(R == v.R);
        Vector<T> ans(R);
        for (std::size_t i = 0; i < R; i++) { ans[i] = table[i] - v[i]; }
        return ans;
    }
    Vector<T> operator*(const T& t) const
    {
        Vector<T> ans(R);
        for (std::size_t i = 0; i < R; i++) { ans[i] = table[i] * t; }
        return ans;
    }
    Vector<T>& operator=(const Vector<T>& v)
    {
        assert(R == v.R);
        for (std::size_t i = 0; i < R; i++) { table[i] = v[i]; }
        return *this;
    }
    Vector<T>& operator+=(const Vector<T>& v)
    {
        assert(R == v.R);
        for (std::size_t i = 0; i < R; i++) { table[i] += v[i]; }
        return *this;
    }
    Vector<T>& operator-=(const Vector<T>& v)
    {
        assert(R == v.R);
        for (std::size_t i = 0; i < R; i++) { table[i] -= v[i]; }
        return *this;
    }
    Vector<T>& operator*=(const T& t)
    {
        Vector<T> ans(R);
        for (std::size_t i = 0; i < R; i++) { table[i] *= t; }
        return *this;
    }
    const std::size_t R;
    std::vector<T> table;
};
template <typename T>
Vector<T> operator*(const T& t, const Vector<T>& v) { return v * t; }
template <typename T>
Vector<T> operator*(const Matrix<T>& m, const Vector<T>& v)
{
    assert(m.C == v.R);
    Vector<T> ans(m.R);
    for (std::size_t i = 0; i < m.R; i++) {
        for (std::size_t j = 0; j < m.C; j++) { ans[i] += m[i][j] * v[j]; }
    }
    return ans;
}
template <typename T>
std::ostream& operator<<(std::ostream& os, const Vector<T>& v)
{
    os << "[\n";
    for (std::size_t i = 0; i < v.R; i++) { os << v[i] << "\n"; }
    return (os << "]\n");
}
template <typename T>
struct TVector
{
    TVector(const std::size_t C) : C{C}, table(C, T(0)) {}
    const T& operator[](const std::size_t c) const { return table[c]; }
    T& operator[](const std::size_t c) { return table[c]; }
    TVector<T> operator+(const TVector<T>& v) const
    {
        assert(C == v.C);
        TVector<T> ans(C);
        for (std::size_t i = 0; i < C; i++) { ans[i] = table[i] + v[i]; }
        return ans;
    }
    TVector<T> operator-(const TVector<T>& v) const
    {
        assert(C == v.C);
        TVector<T> ans(C);
        for (std::size_t i = 0; i < C; i++) { ans[i] = table[i] - v[i]; }
        return ans;
    }
    TVector<T> operator*(const T& t) const
    {
        TVector<T> ans(C);
        for (std::size_t i = 0; i < C; i++) { ans[i] = table[i] * t; }
        return ans;
    }
    TVector<T>& operator=(const TVector<T>& v)
    {
        assert(C == v.C);
        for (std::size_t i = 0; i < C; i++) { table[i] = v[i]; }
        return *this;
    }
    TVector<T>& operator+=(const TVector<T>& v)
    {
        assert(C == v.C);
        for (std::size_t i = 0; i < C; i++) { table[i] += v[i]; }
        return *this;
    }
    TVector<T>& operator-=(const TVector<T>& v)
    {
        assert(C == v.C);
        for (std::size_t i = 0; i < C; i++) { table[i] -= v[i]; }
        return *this;
    }
    TVector<T>& operator*=(const T& t)
    {
        TVector<T> ans(C);
        for (std::size_t i = 0; i < C; i++) { table[i] *= t; }
        return *this;
    }
    const std::size_t C;
    std::vector<T> table;
};
template <typename T>
TVector<T> operator*(const T& t, const TVector<T>& v) { return v * t; }
template <typename T>
TVector<T> operator*(const TVector<T>& v, const Matrix<T>& m)
{
    assert(v.C == m.R);
    TVector<T> ans(m.C);
    for (std::size_t i = 0; i < m.C; i++) {
        for (std::size_t j = 0; j < m.R; j++) { ans[i] += v[j] * m[j][i]; }
    }
    return ans;
}

template <typename T>
std::ostream& operator<<(std::ostream& os, const TVector<T>& v)
{
    os << "[ ";
    for (std::size_t i = 0; i < v.C; i++) { os << v[i] << " "; }
    return (os << "]\n");
}
//!============================================!//
//!    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 mll = ModInt<ll, MOD<ll>>;
using Mat = Matrix<mll>;
int main()
{
    int N;
    ll C;
    std::cin >> N >> C;
    std::vector<mll> a(N);
    for (int i = 0; i < N; i++) { std::cin >> a[i]; }
    Mat m(N, N);
    for (int i = 0; i < N; i++) {
        for (int j = 0; j <= i; j++) { m[i][j] = a[j]; }
    }
    m ^= C;
    mll ans = 0;
    for (int i = 0; i < N; i++) { ans += m[N - 1][i] - (a[i] ^ C); }
    std::cout << ans << std::endl;
    return 0;
}