#include <bits/stdc++.h>
using i32 = int;
using u32 = unsigned int;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
using f64 = double;
using f80 = long double;
using f128 = __float128;
constexpr i32 operator"" _i32(u64 v) { return v; }
constexpr u32 operator"" _u32(u64 v) { return v; }
constexpr i64 operator"" _i64(u64 v) { return v; }
constexpr u64 operator"" _u64(u64 v) { return v; }
constexpr f64 operator"" _f64(f80 v) { return v; }
constexpr f80 operator"" _f80(f80 v) { return v; }
using Istream = std::istream;
using Ostream = std::ostream;
using Str = std::string;
template<typename T>
using Lt = std::less<T>;
template<typename T>
using Gt = std::greater<T>;
template<int n>
using BSet = std::bitset<n>;
template<typename T1, typename T2>
using Pair = std::pair<T1, T2>;
template<typename... Ts>
using Tup = std::tuple<Ts...>;
template<typename T, int N>
using Arr = std::array<T, N>;
template<typename... Ts>
using Deq = std::deque<Ts...>;
template<typename... Ts>
using Set = std::set<Ts...>;
template<typename... Ts>
using MSet = std::multiset<Ts...>;
template<typename... Ts>
using USet = std::unordered_set<Ts...>;
template<typename... Ts>
using UMSet = std::unordered_multiset<Ts...>;
template<typename... Ts>
using Map = std::map<Ts...>;
template<typename... Ts>
using MMap = std::multimap<Ts...>;
template<typename... Ts>
using UMap = std::unordered_map<Ts...>;
template<typename... Ts>
using UMMap = std::unordered_multimap<Ts...>;
template<typename... Ts>
using Vec = std::vector<Ts...>;
template<typename... Ts>
using Stack = std::stack<Ts...>;
template<typename... Ts>
using Queue = std::queue<Ts...>;
template<typename T>
using MaxHeap = std::priority_queue<T>;
template<typename T>
using MinHeap = std::priority_queue<T, Vec<T>, Gt<T>>;
constexpr bool LOCAL = false;
constexpr bool OJ = not LOCAL;
template<typename T>
static constexpr T OjLocal(T oj, T local)
{
    return LOCAL ? local : oj;
}
template<typename T>
constexpr T LIMMIN = std::numeric_limits<T>::min();
template<typename T>
constexpr T LIMMAX = std::numeric_limits<T>::max();
template<typename T>
constexpr T INF = (LIMMAX<T> - 1) / 2;
template<typename T>
constexpr T PI = T{3.141592653589793238462643383279502884};
template<typename T = u64>
constexpr T TEN(int n)
{
    return n == 0 ? T{1} : TEN<T>(n - 1) * T{10};
}
template<typename T>
constexpr bool chmin(T& a, const T& b)
{
    return (a > b ? (a = b, true) : false);
}
template<typename T>
constexpr bool chmax(T& a, const T& b)
{
    return (a < b ? (a = b, true) : false);
}
template<typename T>
constexpr T floorDiv(T x, T y)
{
    assert(y != 0);
    if (y < 0) { x = -x, y = -y; }
    return x >= 0 ? x / y : (x - y + 1) / y;
}
template<typename T>
constexpr T ceilDiv(T x, T y)
{
    assert(y != 0);
    if (y < 0) { x = -x, y = -y; }
    return x >= 0 ? (x + y - 1) / y : x / y;
}
template<typename T, typename I>
constexpr T powerMonoid(T v, I n, const T& e)
{
    assert(n >= 0);
    if (n == 0) { return e; }
    return (n % 2 == 1 ? v * powerMonoid(v, n - 1, e) : powerMonoid(v * v, n / 2, e));
}
template<typename T, typename I>
constexpr T powerInt(T v, I n)
{
    return powerMonoid(v, n, T{1});
}
template<typename Vs, typename V>
constexpr void fillAll(Vs& arr, const V& v)
{
    if constexpr (std::is_convertible<V, Vs>::value) {
        arr = v;
    } else {
        for (auto& subarr : arr) { fillAll(subarr, v); }
    }
}
template<typename Vs>
constexpr void sortAll(Vs& vs)
{
    std::sort(std::begin(vs), std::end(vs));
}
template<typename Vs, typename C>
constexpr void sortAll(Vs& vs, C comp)
{
    std::sort(std::begin(vs), std::end(vs), comp);
}
template<typename Vs>
constexpr void reverseAll(Vs& vs)
{
    std::reverse(std::begin(vs), std::end(vs));
}
template<typename Vs>
constexpr Vs reversed(const Vs& vs)
{
    auto rvs = vs;
    reverseAll(rvs);
    return rvs;
}
template<typename V, typename Vs>
constexpr V sumAll(const Vs& vs)
{
    if constexpr (std::is_convertible<Vs, V>::value) {
        return static_cast<V>(vs);
    } else {
        V ans = 0;
        for (const auto& v : vs) { ans += sumAll<V>(v); }
        return ans;
    }
}
template<typename Vs>
constexpr int minInd(const Vs& vs)
{
    return std::min_element(std::begin(vs), std::end(vs)) - std::begin(vs);
}
template<typename Vs>
constexpr int maxInd(const Vs& vs)
{
    return std::max_element(std::begin(vs), std::end(vs)) - std::begin(vs);
}
template<typename Vs, typename V>
constexpr int lbInd(const Vs& vs, const V& v)
{
    return std::lower_bound(std::begin(vs), std::end(vs), v) - std::begin(vs);
}
template<typename Vs, typename V>
constexpr int ubInd(const Vs& vs, const V& v)
{
    return std::upper_bound(std::begin(vs), std::end(vs), v) - std::begin(vs);
}
template<typename Vs, typename V>
constexpr void plusAll(Vs& vs, const V& v)
{
    for (auto& v_ : vs) { v_ += v; }
}
template<typename Vs>
constexpr void concat(Vs& vs1, const Vs& vs2)
{
    std::copy(std::begin(vs2), std::end(vs2), std::back_inserter(vs1));
}
template<typename Vs>
constexpr void concatted(const Vs& vs1, const Vs& vs2)
{
    auto vs = vs1;
    concat(vs, vs2);
    return vs;
}
template<typename T, typename F>
constexpr Vec<T> genVec(int n, F gen)
{
    Vec<T> ans;
    std::generate_n(std::back_inserter(ans), n, gen);
    return ans;
}
template<typename T = int>
constexpr Vec<T> iotaVec(int n, T offset = 0)
{
    Vec<T> ans(n);
    std::iota(std::begin(ans), std::end(ans), offset);
    return ans;
}
template<typename Vs>
constexpr void rearrange(Vs& vs, const Vec<int>& is)
{
    auto vs_ = vs;
    for (int i = 0; i < (int)is.size(); i++) { vs[i] = vs_[is[i]]; }
}
inline Vec<int> reversePerm(const Vec<int>& is)
{
    auto ris = is;
    for (int i = 0; i < (int)is.size(); i++) { ris[is[i]] = i; }
    return ris;
}
inline Ostream& operator<<(Ostream& os, i128 v)
{
    bool minus = false;
    if (v < 0) { minus = true, v = -v; }
    Str ans;
    if (v == 0) { ans = "0"; }
    while (v) { ans.push_back('0' + v % 10), v /= 10; }
    std::reverse(ans.begin(), ans.end());
    return os << (minus ? "-" : "") << ans;
}
inline Ostream& operator<<(Ostream& os, u128 v)
{
    Str ans;
    if (v == 0) { ans = "0"; }
    while (v) { ans.push_back('0' + v % 10), v /= 10; }
    std::reverse(ans.begin(), ans.end());
    return os << ans;
}
constexpr int popCount(u64 v) { return v ? __builtin_popcountll(v) : 0; }
constexpr int topBit(u64 v) { return v == 0 ? -1 : 63 - __builtin_clzll(v); }
constexpr int lowBit(u64 v) { return v == 0 ? 64 : __builtin_ctzll(v); }
constexpr int bitWidth(u64 v) { return topBit(v) + 1; }
constexpr u64 bitCeil(u64 v) { return v ? (1_u64 << bitWidth(v - 1)) : 1_u64; }
constexpr u64 bitFloor(u64 v) { return v ? (1_u64 << topBit(v)) : 0_u64; }
constexpr bool hasSingleBit(u64 v) { return (v > 0) and ((v & (v - 1)) == 0); }
constexpr bool isBitOn(u64 mask, int ind) { return (mask >> ind) & 1_u64; }
constexpr bool isBitOff(u64 mask, int ind) { return not isBitOn(mask, ind); }
constexpr u64 bitMask(int bitWidth) { return (bitWidth == 64 ? ~0_u64 : (1_u64 << bitWidth) - 1); }
constexpr u64 bitMask(int start, int end) { return bitMask(end - start) << start; }
template<typename F>
struct Fix : F
{
    constexpr Fix(F&& f) : F{std::forward<F>(f)} {}
    template<typename... Args>
    constexpr auto operator()(Args&&... args) const
    {
        return F::operator()(*this, std::forward<Args>(args)...);
    }
};
class irange
{
private:
    struct itr
    {
        constexpr itr(i64 start = 0, i64 step = 1) : m_cnt{start}, m_step{step} {}
        constexpr bool operator!=(const itr& it) const { return m_cnt != it.m_cnt; }
        constexpr i64 operator*() { return m_cnt; }
        constexpr itr& operator++() { return m_cnt += m_step, *this; }
        i64 m_cnt, m_step;
    };
    i64 m_start, m_end, m_step;
public:
    static constexpr i64 cnt(i64 start, i64 end, i64 step)
    {
        if (step == 0) { return -1; }
        const i64 d = (step > 0 ? step : -step);
        const i64 l = (step > 0 ? start : end);
        const i64 r = (step > 0 ? end : start);
        i64 n = (r - l) / d + ((r - l) % d ? 1 : 0);
        if (l >= r) { n = 0; }
        return n;
    }
    constexpr irange(i64 start, i64 end, i64 step = 1)
        : m_start{start}, m_end{m_start + step * cnt(start, end, step)}, m_step{step}
    {
        assert(step != 0);
    }
    constexpr itr begin() const { return itr{m_start, m_step}; }
    constexpr itr end() const { return itr{m_end, m_step}; }
};
constexpr irange rep(i64 end) { return irange(0, end, 1); }
constexpr irange per(i64 rend) { return irange(rend - 1, -1, -1); }
class Scanner
{
public:
    Scanner(Istream& is = std::cin) : m_is{is} { m_is.tie(nullptr)->sync_with_stdio(false); }
    template<typename T>
    T val()
    {
        T v;
        return m_is >> v, v;
    }
    template<typename T>
    T val(T offset)
    {
        return val<T>() - offset;
    }
    template<typename T>
    Vec<T> vec(int n)
    {
        return genVec<T>(n, [&]() { return val<T>(); });
    }
    template<typename T>
    Vec<T> vec(int n, T offset)
    {
        return genVec<T>(n, [&]() { return val<T>(offset); });
    }
    template<typename T>
    Vec<Vec<T>> vvec(int n, int m)
    {
        return genVec<Vec<T>>(n, [&]() { return vec<T>(m); });
    }
    template<typename T>
    Vec<Vec<T>> vvec(int n, int m, const T offset)
    {
        return genVec<Vec<T>>(n, [&]() { return vec<T>(m, offset); });
    }
    template<typename... Args>
    auto tup()
    {
        return Tup<Args...>{val<Args>()...};
    }
    template<typename... Args>
    auto tup(const Args&... offsets)
    {
        return Tup<Args...>{val<Args>(offsets)...};
    }
private:
    Istream& m_is;
};
inline Scanner in;
class Printer
{
public:
    Printer(Ostream& os = std::cout) : m_os{os} { m_os << std::fixed << std::setprecision(15); }
    template<typename... Args>
    int operator()(const Args&... args)
    {
        return dump(args...), 0;
    }
    template<typename... Args>
    int ln(const Args&... args)
    {
        return dump(args...), m_os << '\n', 0;
    }
    template<typename... Args>
    int el(const Args&... args)
    {
        return dump(args...), m_os << std::endl, 0;
    }
    int YES(bool b = true) { return ln(b ? "YES" : "NO"); }
    int NO(bool b = true) { return YES(not b); }
    int Yes(bool b = true) { return ln(b ? "Yes" : "No"); }
    int No(bool b = true) { return Yes(not b); }
private:
    template<typename T>
    void dump(const T& v)
    {
        m_os << v;
    }
    template<typename T>
    void dump(const Vec<T>& vs)
    {
        for (int i : rep(vs.size())) { m_os << (i ? " " : ""), dump(vs[i]); }
    }
    template<typename T>
    void dump(const Vec<Vec<T>>& vss)
    {
        for (int i : rep(vss.size())) { m_os << (i ? "\n" : ""), dump(vss[i]); }
    }
    template<typename T, typename... Ts>
    int dump(const T& v, const Ts&... args)
    {
        return dump(v), m_os << ' ', dump(args...), 0;
    }
    Ostream& m_os;
};
inline Printer out;
template<typename T, int n, int i = 0>
auto ndVec(int const (&szs)[n], const T x = T{})
{
    if constexpr (i == n) {
        return x;
    } else {
        return std::vector(szs[i], ndVec<T, n, i + 1>(szs, x));
    }
}
template<typename T, typename F>
inline T binSearch(T ng, T ok, F check)
{
    while (std::abs(ok - ng) > 1) {
        const T mid = (ok + ng) / 2;
        (check(mid) ? ok : ng) = mid;
    }
    return ok;
}
template<typename T>
constexpr Pair<T, T> extgcd(const T a, const T b) // [x,y] -> ax+by=gcd(a,b)
{
    static_assert(std::is_signed_v<T>, "Signed integer is allowed.");
    assert(a != 0 or b != 0);
    if (a >= 0 and b >= 0) {
        if (a < b) {
            const auto [y, x] = extgcd(b, a);
            return {x, y};
        }
        if (b == 0) { return {1, 0}; }
        const auto [x, y] = extgcd(b, a % b);
        return {y, x - (a / b) * y};
    } else {
        auto [x, y] = extgcd(std::abs(a), std::abs(b));
        if (a < 0) { x = -x; }
        if (b < 0) { y = -y; }
        return {x, y};
    }
}
template<typename T>
constexpr T inverse(const T a, const T mod) // ax=gcd(a,M) (mod M)
{
    assert(a > 0 and mod > 0);
    auto [x, y] = extgcd(a, mod);
    if (x <= 0) { x += mod; }
    return x;
}
template<u32 mod_, u32 root_, u32 max2p_>
class modint
{
    template<typename U = u32&>
    static U modRef()
    {
        static u32 s_mod = 0;
        return s_mod;
    }
    template<typename U = u32&>
    static U rootRef()
    {
        static u32 s_root = 0;
        return s_root;
    }
    template<typename U = u32&>
    static U max2pRef()
    {
        static u32 s_max2p = 0;
        return s_max2p;
    }
public:
    static_assert(mod_ <= LIMMAX<i32>, "mod(signed int size) only supported!");
    static constexpr bool isDynamic() { return (mod_ == 0); }
    template<typename U = const u32>
    static constexpr std::enable_if_t<mod_ != 0, U> mod()
    {
        return mod_;
    }
    template<typename U = const u32>
    static std::enable_if_t<mod_ == 0, U> mod()
    {
        return modRef();
    }
    template<typename U = const u32>
    static constexpr std::enable_if_t<mod_ != 0, U> root()
    {
        return root_;
    }
    template<typename U = const u32>
    static std::enable_if_t<mod_ == 0, U> root()
    {
        return rootRef();
    }
    template<typename U = const u32>
    static constexpr std::enable_if_t<mod_ != 0, U> max2p()
    {
        return max2p_;
    }
    template<typename U = const u32>
    static std::enable_if_t<mod_ == 0, U> max2p()
    {
        return max2pRef();
    }
    template<typename U = u32>
    static void setMod(std::enable_if_t<mod_ == 0, U> m)
    {
        assert(1 <= m and m <= LIMMAX<i32>);
        modRef() = m;
        sinvRef() = {1, 1};
        factRef() = {1, 1};
        ifactRef() = {1, 1};
    }
    template<typename U = u32>
    static void setRoot(std::enable_if_t<mod_ == 0, U> r)
    {
        rootRef() = r;
    }
    template<typename U = u32>
    static void setMax2p(std::enable_if_t<mod_ == 0, U> m)
    {
        max2pRef() = m;
    }
    constexpr modint() : m_val{0} {}
    constexpr modint(i64 v) : m_val{normll(v)} {}
    constexpr void setRaw(u32 v) { m_val = v; }
    constexpr modint operator-() const { return modint{0} - (*this); }
    constexpr modint& operator+=(const modint& m)
    {
        m_val = norm(m_val + m.val());
        return *this;
    }
    constexpr modint& operator-=(const modint& m)
    {
        m_val = norm(m_val + mod() - m.val());
        return *this;
    }
    constexpr modint& operator*=(const modint& m)
    {
        m_val = normll((i64)m_val * (i64)m.val() % (i64)mod());
        return *this;
    }
    constexpr modint& operator/=(const modint& m) { return *this *= m.inv(); }
    constexpr modint operator+(const modint& m) const
    {
        auto v = *this;
        return v += m;
    }
    constexpr modint operator-(const modint& m) const
    {
        auto v = *this;
        return v -= m;
    }
    constexpr modint operator*(const modint& m) const
    {
        auto v = *this;
        return v *= m;
    }
    constexpr modint operator/(const modint& m) const
    {
        auto v = *this;
        return v /= m;
    }
    constexpr bool operator==(const modint& m) const { return m_val == m.val(); }
    constexpr bool operator!=(const modint& m) const { return not(*this == m); }
    friend Istream& operator>>(Istream& is, modint& m)
    {
        i64 v;
        return is >> v, m = v, is;
    }
    friend Ostream& operator<<(Ostream& os, const modint& m) { return os << m.val(); }
    constexpr u32 val() const { return m_val; }
    template<typename I>
    constexpr modint pow(I n) const
    {
        return powerInt(*this, n);
    }
    constexpr modint inv() const { return inverse<i32>(m_val, mod()); }
    static modint sinv(u32 n)
    {
        auto& is = sinvRef();
        for (u32 i = (u32)is.size(); i <= n; i++) { is.push_back(-is[mod() % i] * (mod() / i)); }
        return is[n];
    }
    static modint fact(u32 n)
    {
        auto& fs = factRef();
        for (u32 i = (u32)fs.size(); i <= n; i++) { fs.push_back(fs.back() * i); }
        return fs[n];
    }
    static modint ifact(u32 n)
    {
        auto& ifs = ifactRef();
        for (u32 i = (u32)ifs.size(); i <= n; i++) { ifs.push_back(ifs.back() * sinv(i)); }
        return ifs[n];
    }
    static modint perm(int n, int k) { return k > n or k < 0 ? modint{0} : fact(n) * ifact(n - k); }
    static modint comb(int n, int k)
    {
        return k > n or k < 0 ? modint{0} : fact(n) * ifact(n - k) * ifact(k);
    }
private:
    static Vec<modint>& sinvRef()
    {
        static Vec<modint> is{1, 1};
        return is;
    }
    static Vec<modint>& factRef()
    {
        static Vec<modint> fs{1, 1};
        return fs;
    }
    static Vec<modint>& ifactRef()
    {
        static Vec<modint> ifs{1, 1};
        return ifs;
    }
    static constexpr u32 norm(u32 x) { return x < mod() ? x : x - mod(); }
    static constexpr u32 normll(i64 x) { return norm(u32(x % (i64)mod() + (i64)mod())); }
    u32 m_val;
};
using modint_1000000007 = modint<1000000007, 5, 1>;
using modint_998244353 = modint<998244353, 3, 23>;
template<int id>
using modint_dynamic = modint<0, 0, id>;
template<typename T = int>
class Graph
{
    struct Edge
    {
        Edge() = default;
        Edge(int i, int t, T c) : id{i}, to{t}, cost{c} {}
        int id;
        int to;
        T cost;
        operator int() const { return to; }
    };
public:
    Graph(int n) : m_v{n}, m_edges(n) {}
    void addEdge(int u, int v, bool bi = false)
    {
        assert(0 <= u and u < m_v);
        assert(0 <= v and v < m_v);
        m_edges[u].emplace_back(m_e, v, 1);
        if (bi) { m_edges[v].emplace_back(m_e, u, 1); }
        m_e++;
    }
    void addEdge(int u, int v, const T& c, bool bi = false)
    {
        assert(0 <= u and u < m_v);
        assert(0 <= v and v < m_v);
        m_edges[u].emplace_back(m_e, v, c);
        if (bi) { m_edges[v].emplace_back(m_e, u, c); }
        m_e++;
    }
    const Vec<Edge>& operator[](const int u) const
    {
        assert(0 <= u and u < m_v);
        return m_edges[u];
    }
    Vec<Edge>& operator[](const int u)
    {
        assert(0 <= u and u < m_v);
        return m_edges[u];
    }
    int v() const { return m_v; }
    int e() const { return m_e; }
    friend Ostream& operator<<(Ostream& os, const Graph& g)
    {
        for (int u : rep(g.v())) {
            for (const auto& [id, v, c] : g[u]) {
                os << "[" << id << "]: ";
                os << u << "->" << v << "(" << c << ")\n";
            }
        }
        return os;
    }
    Vec<T> sizes(int root = 0) const
    {
        const int N = v();
        assert(0 <= root and root < N);
        Vec<T> ss(N, 1);
        Fix([&](auto dfs, int u, int p) -> void {
            for ([[maybe_unused]] const auto& [_temp_name_0, v, c] : m_edges[u]) {
                if (v == p) { continue; }
                dfs(v, u);
                ss[u] += ss[v];
            }
        })(root, -1);
        return ss;
    }
    Vec<T> depths(int root = 0) const
    {
        const int N = v();
        assert(0 <= root and root < N);
        Vec<T> ds(N, 0);
        Fix([&](auto dfs, int u, int p) -> void {
            for ([[maybe_unused]] const auto& [_temp_name_1, v, c] : m_edges[u]) {
                if (v == p) { continue; }
                ds[v] = ds[u] + c;
                dfs(v, u);
            }
        })(root, -1);
        return ds;
    }
    Vec<int> parents(int root = 0) const
    {
        const int N = v();
        assert(0 <= root and root < N);
        Vec<int> ps(N, -1);
        Fix([&](auto dfs, int u, int p) -> void {
            for ([[maybe_unused]] const auto& [_temp_name_2, v, c] : m_edges[u]) {
                if (v == p) { continue; }
                ps[v] = u;
                dfs(v, u);
            }
        })(root, -1);
        return ps;
    }
private:
    int m_v;
    int m_e = 0;
    Vec<Vec<Edge>> m_edges;
};
template<typename T>
Vec<T> divisors(const T n)
{
    Vec<T> head, tail;
    for (T i = 1; i * i <= n; i++) {
        if (n % i == 0) {
            head.push_back(i);
            if (i * i != n) { tail.push_back(n / i); }
        }
    }
    reverseAll(tail);
    concat(head, tail);
    return head;
}
template<typename Engine>
class RNG
{
public:
    using result_type = typename Engine::result_type;
    using T = result_type;
    static constexpr T min() { return Engine::min(); }
    static constexpr T max() { return Engine::max(); }
    RNG() : RNG(std::random_device{}()) {}
    RNG(T seed) : m_rng(seed) {}
    T operator()() { return m_rng(); }
    template<typename T>
    T val(T min, T max)
    {
        return std::uniform_int_distribution<T>(min, max)(m_rng);
    }
    template<typename T, typename... Args>
    auto tup(T min, T max, const Args&... offsets)
    {
        return Tup<T, Args...>{val<T>(min, max), val<Args>(offsets)...};
    }
    template<typename T>
    Vec<T> vec(int n, T min, T max)
    {
        return genVec<T>(n, [&]() { return val<T>(min, max); });
    }
    template<typename T>
    Vec<Vec<T>> vvec(int n, int m, T min, T max)
    {
        return genVec<Vec<T>>(n, [&]() { return vec(m, min, max); });
    }
private:
    Engine m_rng;
};
inline RNG<std::mt19937> rng;
inline RNG<std::mt19937_64> rng64;
template<u64 mod_, u64 root_, u64 max2p_>
class modint64
{
    template<typename U = u64&>
    static U modRef()
    {
        static u64 s_mod = 0;
        return s_mod;
    }
    template<typename U = u64&>
    static U rootRef()
    {
        static u64 s_root = 0;
        return s_root;
    }
    template<typename U = u64&>
    static U max2pRef()
    {
        static u64 s_max2p = 0;
        return s_max2p;
    }
public:
    static_assert(mod_ <= LIMMAX<i64>, "mod(signed int size) only supported!");
    static constexpr bool isDynamic() { return (mod_ == 0); }
    template<typename U = const u64>
    static constexpr std::enable_if_t<mod_ != 0, U> mod()
    {
        return mod_;
    }
    template<typename U = const u64>
    static std::enable_if_t<mod_ == 0, U> mod()
    {
        return modRef();
    }
    template<typename U = const u64>
    static constexpr std::enable_if_t<mod_ != 0, U> root()
    {
        return root_;
    }
    template<typename U = const u64>
    static std::enable_if_t<mod_ == 0, U> root()
    {
        return rootRef();
    }
    template<typename U = const u64>
    static constexpr std::enable_if_t<mod_ != 0, U> max2p()
    {
        return max2p_;
    }
    template<typename U = const u64>
    static std::enable_if_t<mod_ == 0, U> max2p()
    {
        return max2pRef();
    }
    template<typename U = u64>
    static void setMod(std::enable_if_t<mod_ == 0, U> m)
    {
        assert(1 <= m and m <= LIMMAX<i64>);
        modRef() = m;
        sinvRef() = {1, 1};
        factRef() = {1, 1};
        ifactRef() = {1, 1};
    }
    template<typename U = u64>
    static void setRoot(std::enable_if_t<mod_ == 0, U> r)
    {
        rootRef() = r;
    }
    template<typename U = u64>
    static void setMax2p(std::enable_if_t<mod_ == 0, U> m)
    {
        max2pRef() = m;
    }
    constexpr modint64() : m_val{0} {}
    constexpr modint64(const i64 v) : m_val{normLL(v)} {}
    constexpr void setRaw(const u64 v) { m_val = v; }
    constexpr modint64 operator+() const { return *this; }
    constexpr modint64 operator-() const { return modint64{0} - (*this); }
    constexpr modint64& operator+=(const modint64& m)
    {
        m_val = norm(m_val + m.val());
        return *this;
    }
    constexpr modint64& operator-=(const modint64& m)
    {
        m_val = norm(m_val + mod() - m.val());
        return *this;
    }
    constexpr modint64& operator*=(const modint64& m)
    {
        m_val = normLL((i128)m_val * (i128)m.val() % (i128)mod());
        return *this;
    }
    constexpr modint64& operator/=(const modint64& m) { return *this *= m.inv(); }
    constexpr modint64 operator+(const modint64& m) const
    {
        auto v = *this;
        return v += m;
    }
    constexpr modint64 operator-(const modint64& m) const
    {
        auto v = *this;
        return v -= m;
    }
    constexpr modint64 operator*(const modint64& m) const
    {
        auto v = *this;
        return v *= m;
    }
    constexpr modint64 operator/(const modint64& m) const
    {
        auto v = *this;
        return v /= m;
    }
    constexpr bool operator==(const modint64& m) const { return m_val == m.val(); }
    constexpr bool operator!=(const modint64& m) const { return not(*this == m); }
    friend Istream& operator>>(Istream& is, modint64& m)
    {
        i64 v;
        return is >> v, m = v, is;
    }
    friend Ostream& operator<<(Ostream& os, const modint64& m) { return os << m.val(); }
    constexpr u64 val() const { return m_val; }
    template<typename I>
    constexpr modint64 pow(I n) const
    {
        return powerInt(*this, n);
    }
    constexpr modint64 inv() const { return inverse<i64>(m_val, mod()); }
    modint64 sinv() const { return sinv(m_val); }
    static modint64 sinv(u32 n)
    {
        auto& is = sinvRef();
        for (u32 i = (u32)is.size(); i <= n; i++) { is.push_back(-is[mod() % i] * (mod() / i)); }
        return is[n];
    }
    static modint64 fact(u32 n)
    {
        auto& fs = factRef();
        for (u32 i = (u32)fs.size(); i <= n; i++) { fs.push_back(fs.back() * i); }
        return fs[n];
    }
    static modint64 ifact(u32 n)
    {
        auto& ifs = ifactRef();
        for (u32 i = (u32)ifs.size(); i <= n; i++) { ifs.push_back(ifs.back() * sinv(i)); }
        return ifs[n];
    }
    static modint64 perm(int n, int k)
    {
        return k > n or k < 0 ? modint64{0} : fact(n) * ifact(n - k);
    }
    static modint64 comb(int n, int k)
    {
        return k > n or k < 0 ? modint64{0} : fact(n) * ifact(n - k) * ifact(k);
    }
private:
    static Vec<modint64>& sinvRef()
    {
        static Vec<modint64> is{1, 1};
        return is;
    }
    static Vec<modint64>& factRef()
    {
        static Vec<modint64> fs{1, 1};
        return fs;
    }
    static Vec<modint64>& ifactRef()
    {
        static Vec<modint64> ifs{1, 1};
        return ifs;
    }
    static constexpr u64 norm(const u64 x) { return x < mod() ? x : x - mod(); }
    static constexpr u64 normLL(const i64 x)
    {
        return norm(u64((i128)x % (i128)mod() + (i128)mod()));
    }
    u64 m_val;
};
template<int id>
using modint64_dynamic = modint64<0, 0, id>;
template<typename mint>
bool millerRabin(u64 n, const Vec<u64>& as)
{
    auto d = n - 1;
    for (; (d & 1) == 0; d >>= 1) {}
    for (const u64 a : as) {
        if (n <= a) { break; }
        auto s = d;
        mint x = mint(a).pow(s);
        while (x.val() != 1 and x.val() != n - 1 and s != n - 1) { x *= x, s <<= 1; }
        if (x.val() != n - 1 and s % 2 == 0) { return false; }
    }
    return true;
}
inline bool isPrime(u64 n)
{
    using mint = modint_dynamic<873293817>;
    using mint64 = modint64_dynamic<828271328>;
    if (n == 1) { return false; }
    if ((n & 1) == 0) { return n == 2; }
    if (n < (1ULL << 30)) {
        mint::setMod(n);
        return millerRabin<mint>(n, {2, 7, 61});
    } else {
        mint64::setMod(n);
        return millerRabin<mint64>(n, {2, 325, 9375, 28178, 450775, 9780504});
    }
}
template<typename mint>
u64 pollardRho(u64 n)
{
    if (n % 2 == 0) { return 2; }
    if (isPrime(n)) { return n; }
    mint c;
    auto f = [&](const mint& x) { return x * x + c; };
    while (true) {
        mint x, y, ys, q = 1;
        y = rng64.val<u64>(0, n - 2) + 2;
        c = rng64.val<u64>(0, n - 2) + 2;
        u64 d = 1;
        constexpr u32 dk = 128;
        for (u32 r = 1; d == 1; r <<= 1) {
            x = y;
            for (u32 i = 0; i < r; i++) { y = f(y); }
            for (u32 k = 0; k < r and d == 1; k += dk) {
                ys = y;
                for (u32 i = 0; i < dk and i < r - k; i++) { q *= x - (y = f(y)); }
                d = std::gcd((u64)q.val(), n);
            }
        }
        if (d == n) {
            do {
                d = std::gcd(u64((x - (ys = f(ys))).val()), n);
            } while (d == 1);
        }
        if (d != n) { return d; }
    }
    return n;
}
Map<u64, int> primeFactors(u64 n)
{
    using mint = modint_dynamic<287687412>;
    using mint64 = modint64_dynamic<4832432>;
    Map<u64, int> ans;
    Fix([&](auto dfs, u64 x) -> void {
        while ((x & 1) == 0) { x >>= 1, ans[2]++; }
        if (x == 1) { return; }
        u64 p;
        if (x < (1ULL << 30)) {
            mint::setMod(x);
            p = pollardRho<mint>(x);
        } else {
            mint64::setMod(x);
            p = pollardRho<mint64>(x);
        }
        if (p == x) {
            ans[p]++;
            return;
        }
        dfs(p), dfs(x / p);
    })(n);
    return ans;
}
Vec<u64> divisors(const u64 n)
{
    const auto fs = primeFactors(n);
    Vec<u64> ds{1};
    for (const auto& [p, e] : fs) {
        u64 pe = p;
        const u32 dn = ds.size();
        for (i32 i = 0; i < e; i++, pe *= p) {
            for (u32 j = 0; j < dn; j++) { ds.push_back(ds[j] * pe); }
        }
    }
    return ds;
}
int main()
{
    using mint = modint_998244353;
    const auto [N, M] = in.tup<int, mint>();
    const auto As = in.vec<i64>(N);
    i64 L = 1;
    for (int i : rep(N)) {
        L = std::lcm(L, As[i]);
    }
    Vec<i64> ps;
    for (const auto& p : primeFactors(L)) {
        ps.push_back(p.first);
    }
    const auto ds = divisors(L);
    mint ans = 0;
    for (const i64 g : ds) {
        i64 ind = 0;
        for (int i : rep(N)) {
            const i64 p = std::gcd(g, As[i]);
            ind += p;
        }
        const mint way = M.pow(ind);
        i64 x = L / g;
        mint phi = x;
        for (i64 p : ps) {
            if (x % p == 0) {
                phi *= mint(p - 1) / p;
            }
        }
        void(0);
        ans += phi * way;
    }
    void(0);
    ans /= L;
    out.ln(ans);
    return 0;
}