/* #region  header */
#pragma GCC optimize("O3") //コンパイラ最適化用

#ifdef LOCAL
#define _GLIBCXX_DEBUG //配列に[]でアクセス時のエラー表示
#endif
#define _USE_MATH_DEFINES
#include <algorithm>   //sort,二分探索,など
#include <atcoder/all> // CodeForcesの場合ファイルごとに入れる
#include <bitset>      //固定長bit集合
// #include <boost/multiprecision/cpp_dec_float.hpp>
// #include <boost/multiprecision/cpp_int.hpp>
#include <cassert> //assert(p)で，not pのときにエラー
#include <cctype>
#include <chrono> //実行時間計測
#include <climits>
#include <cmath>   //pow,logなど
#include <complex> //複素数
#include <cstdio>
#include <cstring>
#include <deque>
#include <functional> //sortのgreater
#include <iomanip>    //setprecision(浮動小数点の出力の誤差)
#include <ios>        // std::left, std::right
#include <iostream>   //入出力
#include <iterator>   //集合演算(積集合,和集合,差集合など)
#include <map>
#include <numeric> //iota(整数列の生成),gcdとlcm,accumulate
#include <queue>
#include <random>
#include <set>
#include <stack>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <utility> //pair
#include <vector>
using namespace std;
using namespace atcoder;
typedef long long LL;
typedef long double LD;

#define ALL(x) x.begin(), x.end()
const long long INF = numeric_limits<long long>::max() / 4;
const int MOD = 1e9 + 7;
// const int MOD=998244353;
//略記
#define FF first
#define SS second
#define int long long
#define stoi stoll
#define LD long double
#define PII pair<int, int>
#define PB push_back
#define EB emplace_back
#define MP make_pair
#define SZ(x) (int)((x).size())
#define VB vector<bool>
#define VVB vector<vector<bool>>
#define VI vector<int>
#define VVI vector<vector<int>>

#define REP(i, n) for (int i = 0; i < (int)(n); i++)
#define REPD(i, n) for (int i = (int)(n) - (int)1; i >= 0; i--)
#define FOR(i, a, b) for (int i = a; i < (int)(b); i++)
#define FORD(i, a, b) for (int i = (int)(b) - (int)1; i >= (int)a; i--)

const int dx[4] = {0, 1, 0, -1}, dy[4] = {-1, 0, 1, 0};
const int Dx[8] = {0, 1, 1, 1, 0, -1, -1, -1},
          Dy[8] = {-1, -1, 0, 1, 1, 1, 0, -1};

int in() {
    int x;
    cin >> x;
    return x;
}
// https://qiita.com/Lily0727K/items/06cb1d6da8a436369eed#c%E3%81%A7%E3%81%AE%E5%AE%9F%E8%A3%85
void print() { cout << "\n"; }
template <class Head, class... Tail> void print(Head &&head, Tail &&...tail) {
    cout << head;
    if (sizeof...(tail) != 0)
        cout << " ";
    print(forward<Tail>(tail)...);
}
template <class T> void print(vector<T> &vec) {
    for (auto &a : vec) {
        cout << a;
        if (&a != &vec.back())
            cout << " ";
    }
    cout << "\n";
}
template <class T> void print(set<T> &set) {
    for (auto &a : set) {
        cout << a << " ";
    }
    cout << "\n";
}
template <class T> void print(vector<vector<T>> &df) {
    for (auto &vec : df) {
        print(vec);
    }
}

// debug macro
// https://atcoder.jp/contests/abc202/submissions/22815715
namespace debugger {
template <class T> void view(const std::vector<T> &a) {
    std::cerr << "{ ";
    for (const auto &v : a) {
        std::cerr << v << ", ";
    }
    std::cerr << "\b\b }";
}
template <class T> void view(const std::vector<std::vector<T>> &a) {
    std::cerr << "{\n";
    for (const auto &v : a) {
        std::cerr << "\t";
        view(v);
        std::cerr << "\n";
    }
    std::cerr << "}";
}
template <class T, class U> void view(const std::vector<std::pair<T, U>> &a) {
    std::cerr << "{\n";
    for (const auto &p : a)
        std::cerr << "\t(" << p.first << ", " << p.second << ")\n";
    std::cerr << "}";
}
template <class T, class U> void view(const std::map<T, U> &m) {
    std::cerr << "{\n";
    for (const auto &p : m)
        std::cerr << "\t[" << p.first << "] : " << p.second << "\n";
    std::cerr << "}";
}
template <class T, class U> void view(const std::pair<T, U> &p) {
    std::cerr << "(" << p.first << ", " << p.second << ")";
}
template <class T> void view(const std::set<T> &s) {
    std::cerr << "{ ";
    for (auto &v : s) {
        view(v);
        std::cerr << ", ";
    }
    std::cerr << "\b\b }";
}

template <class T> void view(const T &e) { std::cerr << e; }
} // namespace debugger
#ifdef LOCAL
void debug_out() {}
template <typename Head, typename... Tail> void debug_out(Head H, Tail... T) {
    debugger::view(H);
    std::cerr << ", ";
    debug_out(T...);
}
#define debug(...)                                                             \
    do {                                                                       \
        std::cerr << __LINE__ << " [" << #__VA_ARGS__ << "] : [";              \
        debug_out(__VA_ARGS__);                                                \
        std::cerr << "\b\b]\n";                                                \
    } while (false)
#else
#define debug(...) (void(0))
#endif

long long power(long long x, long long n) {
    // O(logn)
    // https://algo-logic.info/calc-pow/#toc_id_1_2
    long long ret = 1;
    while (n > 0) {
        if (n & 1)
            ret *= x; // n の最下位bitが 1 ならば x^(2^i) をかける
        x *= x;
        n >>= 1; // n を1bit 左にずらす
    }
    return ret;
}
// @brief nCr. O(r log n)。あるいは前処理 O(n), 本処理 O(1)で求められる modint
// の bc を検討。
int comb(int n, int r) {
    // https://www.geeksforgeeks.org/program-to-calculate-the-value-of-ncr-efficiently/
    if (n < r)
        return 0;

    // p holds the value of n*(n-1)*(n-2)...,
    // k holds the value of r*(r-1)...
    long long p = 1, k = 1;

    // C(n, r) == C(n, n-r),
    // choosing the smaller value
    if (n - r < r)
        r = n - r;

    if (r != 0) {
        while (r) {
            p *= n;
            k *= r;

            // gcd of p, k
            long long m = __gcd(p, k);

            // dividing by gcd, to simplify
            // product division by their gcd
            // saves from the overflow
            p /= m;
            k /= m;

            n--;
            r--;
        }

        // k should be simplified to 1
        // as C(n, r) is a natural number
        // (denominator should be 1 ) .
    }

    else
        p = 1;

    // if our approach is correct p = ans and k =1
    return p;
}

// MOD
void add(long long &a, long long b) {
    a += b;
    if (a >= MOD)
        a -= MOD;
}
template <class T> inline bool chmin(T &a, T b) {
    if (a > b) {
        a = b;
        return true;
    }
    return false;
}
template <class T> inline bool chmax(T &a, T b) {
    if (a < b) {
        a = b;
        return true;
    }
    return false;
}

// 負数も含む丸め
long long ceildiv(long long a, long long b) {
    if (b < 0)
        a = -a, b = -b;
    if (a >= 0)
        return (a + b - 1) / b;
    else
        return a / b;
}
long long floordiv(long long a, long long b) {
    if (b < 0)
        a = -a, b = -b;
    if (a >= 0)
        return a / b;
    else
        return (a - b + 1) / b;
}
long long floorsqrt(long long x) {
    assert(x >= 0);
    long long ok = 0;
    long long ng = 1;
    while (ng * ng <= x)
        ng <<= 1;
    while (ng - ok > 1) {
        long long mid = (ng + ok) >> 1;
        if (mid * mid <= x)
            ok = mid;
        else
            ng = mid;
    }
    return ok;
}

// @brief a^n mod mod
long long modpower(long long a, long long n, long long mod) {
    long long res = 1;
    while (n > 0) {
        if (n & 1)
            res = res * a % mod;
        a = a * a % mod;
        n >>= 1;
    }
    return res;
}

// @brief s が c を含むか
template <class T> bool contain(const std::string &s, const T &c) {
    return s.find(c) != std::string::npos;
}

__attribute__((constructor)) void faster_io() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);
    cerr.tie(nullptr);
}
/* #endregion */

/* #region Math Formal Power Series */
// 二項係数ライブラリ
template <class T> struct BiCoef {
    vector<T> fact_, inv_, finv_;
    constexpr BiCoef() {}
    constexpr BiCoef(int n) noexcept : fact_(n, 1), inv_(n, 1), finv_(n, 1) {
        init(n);
    }

    // @brief O(n)
    constexpr void init(int n, int mod) noexcept {
        fact_.assign(n, 1), inv_.assign(n, 1), finv_.assign(n, 1);
        int MOD = mod;
        for (int i = 2; i < n; i++) {
            fact_[i] = fact_[i - 1] * i;
            inv_[i] = -inv_[MOD % i] * (MOD / i);
            finv_[i] = finv_[i - 1] * inv_[i];
        }
    }
    // @brief O(1)
    constexpr T com(int n, int k) const noexcept {
        if (n < k || n < 0 || k < 0)
            return 0;
        return fact_[n] * finv_[k] * finv_[n - k];
    }
    // @brief O(1)
    constexpr T fact(int n) const noexcept {
        if (n < 0)
            return 0;
        return fact_[n];
    }
    // @brief O(1)
    constexpr T perm(int n, int k) const noexcept {
        if (n < k || n < 0 || k < 0)
            return 0;
        return fact_[n] * finv_[k] * finv_[n - k] * fact_[k];
    }
    // @brief O(1)
    constexpr T inv(int n) const noexcept {
        if (n < 0)
            return 0;
        return inv_[n];
    }
    // @brief O(1)
    constexpr T finv(int n) const noexcept {
        if (n < 0)
            return 0;
        return finv_[n];
    }
};

enum Mode {
    FAST = 1,
    NAIVE = -1,
};
template <class T, Mode mode = FAST> struct FormalPowerSeries : std::vector<T> {
    using std::vector<T>::vector;
    using std::vector<T>::size;
    using std::vector<T>::resize;
    using std::vector<T>::begin;
    using std::vector<T>::insert;
    using std::vector<T>::erase;
    using F = FormalPowerSeries;
    using S = std::vector<std::pair<int, T>>;

    F &operator+=(const F &g) {
        for (int i = 0; i < (int)(std::min((*this).size(), g.size())); i++)
            (*this)[i] += g[i];
        return *this;
    }

    F &operator+=(const T &t) {
        assert((int)((*this).size()));
        (*this)[0] += t;
        return *this;
    }

    F &operator-=(const F &g) {
        for (int i = 0; i < (int)(std::min((*this).size(), g.size())); i++)
            (*this)[i] -= g[i];
        return *this;
    }

    F &operator-=(const T &t) {
        assert(SZ((*this)));
        (*this)[0] -= t;
        return *this;
    }

    F &operator*=(const T &t) {
        for (int i = 0; i < SZ((*this)); ++i)
            (*this)[i] *= t;
        return *this;
    }

    F &operator/=(const T &t) {
        T div = t.inv();
        for (int i = 0; i < SZ(*this); ++i)
            (*this)[i] *= div;
        return *this;
    }

    F &operator>>=(const int sz) {
        assert(sz >= 0);
        int n = (*this).size();
        (*this).erase((*this).begin(), (*this).begin() + std::min(sz, n));
        (*this).resize(n);
        return *this;
    }

    F &operator<<=(const int sz) {
        assert(sz >= 0);
        int n = (*this).size();
        (*this).insert((*this).begin(), sz, T(0));
        (*this).resize(n);
        return *this;
    }

    F &operator%=(const F &g) { return *this -= *this / g * g; }

    F &operator=(const std::vector<T> &v) {
        int n = (*this).size();
        for (int i = 0; i < n; ++i)
            (*this)[i] = v[i];
        return *this;
    }

    F operator-() const {
        F ret = *this;
        return ret * -1;
    }

    F &operator*=(const F &g) {
        if (mode == FAST) {
            int n = (*this).size();
            auto tmp = atcoder::convolution(*this, g);
            for (int i = 0; i < n; ++i)
                (*this)[i] = tmp[i];
            return *this;
        } else {
            int n = (*this).size(), m = g.size();
            for (int i = n - 1; i >= 0; --i) {
                (*this)[i] *= g[0];
                for (int j = 1; j < std::min(i + 1, m); j++)
                    (*this)[i] += (*this)[i - j] * g[j];
            }
            return *this;
        }
    }

    F &operator/=(const F &g) {
        if ((*this).size() < g.size()) {
            (*this).assign((*this).size(), T(0));
            return *this;
        }
        if (mode == FAST) {
            int old = (*this).size();
            int n = (*this).size() - g.size() + 1;
            *this = ((*this).rev().pre(n) * g.rev().inv(n));
            (*this).rev_inplace();
            (*this).resize(old);
            return *this;
        } else {
            assert(g[0] != T(0));
            T ig0 = g[0].inv();
            int n = (*this).size(), m = g.size();
            for (int i = 0; i < n; ++i) {
                for (int j = 1; j < std::min(i + 1, m); ++j)
                    (*this)[i] -= (*this)[i - j] * g[j];
                (*this)[i] *= ig0;
            }
            return *this;
        }
    }

    F &operator*=(S g) {
        int n = (*this).size();
        auto [d, c] = g.front();
        if (!d)
            g.erase(g.begin());
        else
            c = 0;
        for (int i = n - 1; i >= 0; --i) {
            (*this)[i] *= c;
            for (auto &[j, b] : g) {
                if (j > i)
                    break;
                (*this)[i] += (*this)[i - j] * b;
            }
        }
        return *this;
    }

    F &operator/=(S g) {
        int n = (*this).size();
        auto [d, c] = g.front();
        assert(!d and c != 0);
        T ic = c.inv();
        g.erase(g.begin());
        for (int i = 0; i < n; ++i) {
            for (auto &[j, b] : g) {
                if (j > i)
                    break;
                (*this)[i] -= (*this)[i - j] * b;
            }
            (*this)[i] *= ic;
        }
        return *this;
    }

    F operator+(const F &g) const { return F(*this) += g; }

    F operator+(const T &t) const { return F(*this) += t; }

    F operator-(const F &g) const { return F(*this) -= g; }

    F operator-(const T &t) const { return F(*this) -= t; }

    F operator*(const F &g) const { return F(*this) *= g; }

    F operator*(const T &t) const { return F(*this) *= t; }

    F operator/(const F &g) const { return F(*this) /= g; }

    F operator/(const T &t) const { return F(*this) /= t; }

    F operator%(const F &g) const { return F(*this) %= g; }

    F operator*=(const S &g) const { return F(*this) *= g; }

    F operator/=(const S &g) const { return F(*this) /= g; }

    F pre(int d) const {
        return F((*this).begin(),
                 (*this).begin() + std::min((int)(*this).size(), d));
    }

    F &shrink() {
        while (!(*this).empty() and (*this).back() == T(0))
            (*this).pop_back();
        return *this;
    }

    F &rev_inplace() {
        reverse((*this).begin(), (*this).end());
        return *this;
    }
    F rev() const { return F(*this).rev_inplace(); }

    // *=(1 + cz^d)
    F &multiply(const int d, const T c) {
        int n = (*this).size();
        if (c == T(1))
            for (int i = n - d - 1; i >= 0; --i)
                (*this)[i + d] += (*this)[i];
        else if (c == T(-1))
            for (int i = n - d - 1; i >= 0; --i)
                (*this)[i + d] -= (*this)[i];
        else
            for (int i = n - d - 1; i >= 0; --i)
                (*this)[i + d] += (*this)[i] * c;
        return *this;
    }
    // /=(1 + cz^d)
    F &divide(const int d, const T c) {
        int n = (*this).size();
        if (c == T(1))
            for (int i = 0; i < n - d; ++i)
                (*this)[i + d] -= (*this)[i];
        else if (c == T(-1))
            for (int i = 0; i < n - d; ++i)
                (*this)[i + d] += (*this)[i];
        else
            for (int i = 0; i < n - d; ++i)
                (*this)[i + d] -= (*this)[i] * c;
        return *this;
    }

    // @brief O(N)
    T eval(const T &t) const {
        int n = (*this).size();
        T res = 0, tmp = 1;
        for (int i = 0; i < n; ++i)
            res += (*this)[i] * tmp, tmp *= t;
        return res;
    }

    // @brief O(n log n). FAST のみ
    F inv(int deg = -1) const {
        int n = (*this).size();
        assert(mode == FAST and n and (*this)[0] != 0);
        if (deg == -1)
            deg = n;
        assert(deg > 0);
        F res{(*this)[0].inv()};
        while (SZ(res) < deg) {
            int m = res.size();
            F f((*this).begin(), (*this).begin() + std::min(n, m * 2)), r(res);
            f.resize(m * 2), atcoder::internal::butterfly(f);
            r.resize(m * 2), atcoder::internal::butterfly(r);
            for (int i = 0; i < m * 2; ++i)
                f[i] *= r[i];
            atcoder::internal::butterfly_inv(f);
            f.erase(f.begin(), f.begin() + m);
            f.resize(m * 2), atcoder::internal::butterfly(f);
            for (int i = 0; i < m * 2; ++i)
                f[i] *= r[i];
            atcoder::internal::butterfly_inv(f);
            T iz = T(m * 2).inv();
            iz *= -iz;
            for (int i = 0; i < m; ++i)
                f[i] *= iz;
            res.insert(res.end(), f.begin(), f.begin() + m);
        }
        res.resize(deg);
        return res;
    }

    // @brief Ο(N)
    F &diff_inplace() {
        int n = (*this).size();
        for (int i = 1; i < n; ++i)
            (*this)[i - 1] = (*this)[i] * i;
        (*this)[n - 1] = 0;
        return *this;
    }
    F diff() const { F(*this).diff_inplace(); }

    // @brief Ο(N)
    F &integral_inplace() {
        int n = (*this).size(), mod = T::mod();
        std::vector<T> inv(n);
        {
            inv[1] = 1;
            for (int i = 2; i < n; ++i)
                inv[i] = T(mod) - inv[mod % i] * (mod / i);
        }
        for (int i = n - 2; i >= 0; --i)
            (*this)[i + 1] = (*this)[i] * inv[i + 1];
        (*this)[0] = 0;
        return *this;
    }
    F integral() const { return F(*this).integral_inplace(); }

    // @brief Ο(NlogN). FAST のみ
    F &log_inplace() {
        int n = (*this).size();
        assert(n and (*this)[0] == 1);
        F f_inv = (*this).inv();
        (*this).diff_inplace();
        (*this) *= f_inv;
        (*this).integral_inplace();
        return *this;
    }
    F log() const { return F(*this).log_inplace(); }

    //Ο(NlogN)
    F &deriv_inplace() {
        int n = (*this).size();
        assert(n);
        for (int i = 2; i < n; ++i)
            (*this)[i] *= i;
        (*this).erase((*this).begin());
        (*this).push_back(0);
        return *this;
    }
    F deriv() const { return F(*this).deriv_inplace(); }

    // @brief Ο(NlogN). FAST のみ
    F &exp_inplace() {
        int n = (*this).size();
        assert(n and (*this)[0] == 0);
        F g{1};
        (*this)[0] = 1;
        F h_drv((*this).deriv());
        for (int m = 1; m < n; m *= 2) {
            F f((*this).begin(), (*this).begin() + m);
            f.resize(2 * m), atcoder::internal::butterfly(f);
            auto mult_f = [&](F &p) {
                p.resize(2 * m);
                atcoder::internal::butterfly(p);
                for (int i = 0; i < 2 * m; ++i)
                    p[i] *= f[i];
                atcoder::internal::butterfly_inv(p);
                p /= 2 * m;
            };
            if (m > 1) {
                F g_(g);
                g_.resize(2 * m), atcoder::internal::butterfly(g_);
                for (int i = 0; i < 2 * m; ++i)
                    g_[i] *= g_[i] * f[i];
                atcoder::internal::butterfly_inv(g_);
                T iz = T(-2 * m).inv();
                g_ *= iz;
                g.insert(g.end(), g_.begin() + m / 2, g_.begin() + m);
            }
            F t((*this).begin(), (*this).begin() + m);
            t.deriv_inplace();
            {
                F r{h_drv.begin(), h_drv.begin() + m - 1};
                mult_f(r);
                for (int i = 0; i < m; ++i)
                    t[i] -= r[i] + r[m + i];
            }
            t.insert(t.begin(), t.back());
            t.pop_back();
            t *= g;
            F v((*this).begin() + m, (*this).begin() + std::min(n, 2 * m));
            v.resize(m);
            t.insert(t.begin(), m - 1, 0);
            t.push_back(0);
            t.integral_inplace();
            for (int i = 0; i < m; ++i)
                v[i] -= t[m + i];
            mult_f(v);
            for (int i = 0; i < std::min(n - m, m); ++i)
                (*this)[m + i] = v[i];
        }
        return *this;
    }
    F exp() const { return F(*this).exp_inplace(); }

    // @brief Ο(NlogN). FAST のみ
    F &pow_inplace(long long k) {
        int n = (*this).size(), l = 0;
        assert(k >= 0);
        if (!k) {
            for (int i = 0; i < n; ++i)
                (*this)[i] = !i;
            return *this;
        }
        while (l < n and (*this)[l] == 0)
            ++l;
        if (l > (n - 1) / k or l == n)
            return *this = F(n);
        T c = (*this)[l];
        (*this).erase((*this).begin(), (*this).begin() + l);
        (*this) /= c;
        (*this).log_inplace();
        (*this).resize(n - l * k);
        (*this) *= k;
        (*this).exp_inplace();
        (*this) *= c.pow(k);
        (*this).insert((*this).begin(), l * k, 0);
        return *this;
    }
    // @brief Ο(NlogN). FAST のみ
    F pow(const long long k) const { return F(*this).pow_inplace(k); }

    // @brief Ο(NlogN). FAST のみ
    F sqrt(int deg = -1) const {
        auto SQRT = [&](T t) {
            int mod = T::mod();
            if (t == 0 or t == 1)
                return t;
            int v = (mod - 1) / 2;
            if (t.pow(v) != 1)
                return T(-1);
            int q = mod - 1, m = 0;
            while (~q & 1)
                q >>= 1, m++;
            std::mt19937 mt;
            T z = mt();
            while (z.pow(v) != mod - 1)
                z = mt();
            T c = z.pow(q), u = t.pow(q), r = t.pow((q + 1) / 2);
            for (; m > 1; m--) {
                T tmp = u.pow(1 << (m - 2));
                if (tmp != 1)
                    r = r * c, u = u * c * c;
                c = c * c;
            }
            return T(std::min(r.val(), mod - r.val()));
        };
        int n = (*this).size();
        if (deg == -1)
            deg = n;
        if ((*this)[0] == 0) {
            for (int i = 1; i < n; i++) {
                if ((*this)[i] != 0) {
                    if (i & 1)
                        return F(0);
                    if (deg - i / 2 <= 0)
                        break;
                    auto ret = (*this);
                    ret >>= i;
                    ret.resize(n - i);
                    ret = ret.sqrt(deg - i / 2);
                    if (ret.empty())
                        return F(0);
                    ret <<= (i / 2);
                    ret.resize(deg);
                    return ret;
                }
            }
            return F(deg);
        }
        auto sqr = SQRT((*this)[0]);
        if (sqr * sqr != (*this)[0])
            return F(0);
        F ret{sqr};
        T ti = T(1) / T(2);
        for (int i = 1; i < deg; i <<= 1) {
            auto u = (*this);
            u.resize(i << 1);
            ret = (ret.inv(i << 1) * u + ret) * ti;
        }
        ret.resize(deg);
        return ret;
    }

    // WARNING: TODO
    void sparse_pow(const int n, const int d, const T c, const int k);
    void sparse_pow_inv(const int n, const int d, const T c, const int k);
    void stirling_first(int n);
    void stirling_second(int n);
    std::vector<T> multipoint_evaluation(const std::vector<T> &p);

    // @brief O(_MAX log n)
    F &binomial_neg(int r, int d, int _MAX, BiCoef<T> &bc) {
        REP(n, _MAX) { (*this)[n] = bc.com(n + d - 1, d - 1) * T(r).pow(n); }
        return *this;
    }
};
/* #endregion Math Formal Power Series */

/* #region  barlekamp_massey_and_bostan_mori */
template <class F> F Berlekamp_Massey(const F &a) {
    using T = typename F::value_type;
    int n = a.size();
    F c{-1}, c2{0};
    T r2 = 1;
    int i2 = -1;
    for (int i = 0; i < n; i++) {
        T r = 0;
        int d = c.size();
        for (int j = 0; j < d; j++)
            r += c[j] * a[i - j];
        if (r == 0)
            continue;
        T coef = -r / r2;
        int d2 = c2.size();
        if (d - i >= d2 - i2) {
            for (int j = 0; j < d2; j++)
                c[j + i - i2] += c2[j] * coef;
        } else {
            F tmp(c);
            c.resize(d2 + i - i2);
            for (int j = 0; j < d2; j++)
                c[j + i - i2] += c2[j] * coef;
            c2 = std::move(tmp);
            i2 = i, r2 = r;
        }
    }
    return {c.begin() + 1, c.end()};
}

// return generating function of a, s.t. F(x) = P(x) / Q(x)
template <class F> std::pair<F, F> find_generating_function(F a) {
    auto q = Berlekamp_Massey(a);
    int d = q.size();
    a.resize(d);
    q.insert(q.begin(), 1);
    for (int i = 1; i < (int)q.size(); i++)
        q[i] *= -1;
    a *= q;
    return {a, q};
}

// return [x^k] p(x) / q(x)
template <class T, Mode mode>
T compute_Kthterm(FormalPowerSeries<T, mode> p, FormalPowerSeries<T, mode> q,
                  long long k) {
    int d = q.size();
    assert(q[0] == 1 and p.size() + 1 <= d);
    while (k) {
        auto q_minus = q;
        for (int i = 1; i < d; i += 2)
            q_minus[i] *= -1;
        p.resize(2 * d);
        q.resize(2 * d);
        p *= q_minus;
        q *= q_minus;
        for (int i = 0; i < d - 1; i++)
            p[i] = p[(i << 1) | (k & 1)];
        for (int i = 0; i < d; i++)
            q[i] = q[i << 1];
        p.resize(d - 1);
        q.resize(d);
        k >>= 1;
    }
    return p[0];
}

template <class T, Mode mode>
T compute_Kthterm(
    std::pair<FormalPowerSeries<T, mode>, FormalPowerSeries<T, mode>> f,
    long long k) {
    return compute_Kthterm(f.first, f.second, k);
}
/* #endregion barlekamp_massey_and_bostan_mori */

using mint = modint1000000007;
using fps = FormalPowerSeries<mint, NAIVE>;
int p, q, r, K;
signed main() {
    cin >> p >> q >> r >> K;
    // https://yukicoder.me/submissions/655024
    int n = 100000;
    fps a(n, 1);
    a[0] = p % 10, a[1] = q % 10, a[2] = r % 10;
    FOR(ni, 3, n) {
        a[ni] = (a[ni - 3].val() + a[ni - 2].val() + a[ni - 1].val()) % 10;
    }
    // REP(ni, n) { cerr << a[ni].val() << " "; }
    // debug();
    auto f = find_generating_function(a);
    print(compute_Kthterm(f, K - 1).val() % 10);
}