#include <bits/stdc++.h>
using namespace std;
#include <atcoder/all>
using namespace atcoder;

#define int long long
// #define endl "\n"

#pragma GCC optimize("-O3")

void solve();

typedef long long ll;
typedef __int128_t LL;
typedef unsigned long long ull;
typedef double db;
typedef long double ld;
typedef pair<int, int> pi;
typedef pair<int, pair<int, int>> pip;
typedef vector<int> vi;
typedef vector<double> vd;
typedef vector<bool> vb;
typedef vector<string> vs;
typedef vector<char> vc;
typedef vector<pair<int, int>> vp;
typedef vector<vector<int>> vvi;
typedef vector<vector<double>> vvd;
typedef vector<vector<bool>> vvb;
typedef vector<vector<string>> vvs;
typedef vector<vector<char>> vvc;
typedef vector<vector<pair<int, int>>> vvp;
typedef vector<vector<vector<int>>> vvvi;
typedef vector<vector<vector<vector<int>>>> vvvvi;
template <typename T>
using vec = vector<T>;
template <typename T>
using vv = vector<vector<T>>;
template <typename T>
using vvv = vector<vector<vector<T>>>;
template <typename T>
using vvvv = vector<vector<vector<vector<T>>>>;
template <typename T>
using pq = priority_queue<T>;
template <typename T>
using pqg = priority_queue<T, vector<T>, greater<T>>;
template <typename T>
using mset = multiset<T>;
template <typename T>
using uset = unordered_set<T>;
template <typename T, typename U>
using umap = unordered_map<T, U>;

#define _PI 3.14159265358979323846
#define _E 2.7182818284590452354
#define fi first
#define se second
#define pb push_back
#define eb emplace_back
#define mp make_pair
#define mt make_tuple
#define td typedef
#define elif else if
#define ifnot(x) if(!(x))
#define all(obj) (obj).begin(), (obj).end()
#define rall(obj) (obj).rbegin(), (obj).rend()
#define sumv(a) accumulate(all(a), 0LL)
#define lb(v, a) (lower_bound(begin(v), end(v), a) - begin(v))
#define ub(v, a) (upper_bound(begin(v), end(v), a) - begin(v))
#define inr(l, x, r) (l <= x && x < r)
#define cbit(x) __builtin_popcountll(x)
#define topbit(t) (t == 0 ? -1 : 63 - __builtin_clzll(t))
#define botbit(t) (t == 0 ? 64 : __builtin_ctzll(t))
#define gbit(msk, i) ((msk) >> (i) & 1)
#define mask(x) ((1LL << (x)) - 1)
#define setbits(i, n) \
    for(int j = (n), i = botbit(j); j; j ^= 1LL << i, i = botbit(j))

#define rep1(a)                                                  \
    for(int NEVER_USE_VARIABLE = 0; NEVER_USE_VARIABLE < (int)a; \
        NEVER_USE_VARIABLE++)
#define rep2(i, a) for(int i = 0; i < (int)a; i++)
#define rep3(i, a, b) for(int i = a; i < (int)b; i++)
#define rep4(i, a, b, c) for(int i = a; i < (int)b; i += c)
#define overload4(a, b, c, d, e, ...) e
#define rep(...) overload4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__)
#define rrep1(n) for(ll NEVER_USE_VARIABLE = n; NEVER_USE_VARIABLE--;)
#define rrep2(i, n) for(ll i = n; i--;)
#define rrep3(i, a, b) for(ll i = b; i-- > (a);)
#define rrep4(i, a, b, c) \
    for(ll i = (a) + ((b) - (a)-1) / (c) * (c); i >= (a); i -= c)
#define rrep(...) \
    overload4(__VA_ARGS__, rrep4, rrep3, rrep2, rrep1)(__VA_ARGS__)
#define fore1(i, a) for(auto &&i : a)
#define fore2(x, y, a) for(auto &&[x, y] : a)
#define fore3(x, y, z, a) for(auto &&[x, y, z] : a)
#define fore(...) overload4(__VA_ARGS__, fore3, fore2, fore1)(__VA_ARGS__)
#define ryes return yes();
#define rno return no();
#define rerr return err();

istream &operator>>(istream &is, modint998244353 &a) {
    long long v;
    is >> v;
    a = v;
    return is;
}
ostream &operator<<(ostream &os, const modint998244353 &a) {
    return os << a.val();
}
istream &operator>>(istream &is, modint1000000007 &a) {
    long long v;
    is >> v;
    a = v;
    return is;
}
ostream &operator<<(ostream &os, const modint1000000007 &a) {
    return os << a.val();
}
template <class T, class U>
istream &operator>>(istream &is, pair<T, U> &p) {
    is >> p.first >> p.second;
    return is;
}
template <class T, class U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
    os << p.first << "," << p.second;
    return os;
}
template <class T>
ostream &operator<<(ostream &s, set<T> P) {
    fore(it, P) {
        s << it << " ";
    }
    return s;
}
template <class T1, class T2>
ostream &operator<<(ostream &s, map<T1, T2> P) {
    fore(x, y, P) {
        s << "<" << x << "->" << y << "> ";
    }
    return s;
}
template <class T>
ostream &operator<<(ostream &s, multiset<T> P) {
    fore(it, P) {
        s << it << " ";
    }
    return s;
}
template <class T>
ostream &operator<<(ostream &s, unordered_set<T> P) {
    fore(it, P) {
        s << it << " ";
    }
    return s;
}
template <class T1, class T2>
ostream &operator<<(ostream &s, unordered_map<T1, T2> P) {
    fore(x, y, P) {
        s << "<" << x << "->" << y << "> ";
    }
    return s;
}
template <class T>
istream &operator>>(istream &is, vector<T> &v) {
    for(auto &e : v) is >> e;
    return is;
}
template <class T>
ostream &operator<<(ostream &os, const vector<T> &v) {
    for(auto &e : v) os << e << ' ';
    return os;
}
template <class T>
ostream &operator<<(ostream &os, const vector<vector<T>> &v) {
    for(auto &e : v) {
        for(auto &c : e) os << c << ' ';
        os << endl;
    }
    return os;
}
template <class T>
vector<T> &operator++(vector<T> &v) {
    for(auto &e : v) e++;
    return v;
}
template <class T>
vector<T> operator++(vector<T> &v, signed) {
    auto res = v;
    for(auto &e : v) e++;
    return res;
}
template <class T>
vector<T> &operator--(vector<T> &v) {
    for(auto &e : v) e--;
    return v;
}
template <class T>
vector<T> operator--(vector<T> &v, signed) {
    auto res = v;
    for(auto &e : v) e--;
    return res;
}

// 十進数からb進数へ
template <class T>
string to_baseB(T x, int b = 10) {
    string ans;
    do {
        int num = x % b;
        ans = (char)((num <= 9) ? ('0' + num) : ('A' + num - 10)) + ans;
        x /= b;
    } while(x != 0);
    return ans;
}
// b進数から十進数へ
long long to_base10(const string &x, int b = 10) {
    long long ans = 0, base = 1;
    for(int i = x.length() - 1; i >= 0; --i) {
        int num =
            ('0' <= x[i] && x[i] <= '9') ? (x[i] - '0') : (x[i] - 'A' + 10);
        ans += base * num;
        base *= b;
    }
    return ans;
}

ostream &operator<<(ostream &s, const LL &p) {
    s << to_baseB(p);
    return s;
}

// debug methods
// usage: debug(x,y);
#define CHOOSE(a) CHOOSE2 a
#define CHOOSE2(a0, a1, a2, a3, a4, x, ...) x
#define debug_1(x1) cout << #x1 << ": " << x1 << endl
#define debug_2(x1, x2) \
    cout << #x1 << ": " << x1 << ", " #x2 << ": " << x2 << endl
#define debug_3(x1, x2, x3)                                                 \
    cout << #x1 << ": " << x1 << ", " #x2 << ": " << x2 << ", " #x3 << ": " \
         << x3 << endl
#define debug_4(x1, x2, x3, x4)                                             \
    cout << #x1 << ": " << x1 << ", " #x2 << ": " << x2 << ", " #x3 << ": " \
         << x3 << ", " #x4 << ": " << x4 << endl
#define debug_5(x1, x2, x3, x4, x5)                                         \
    cout << #x1 << ": " << x1 << ", " #x2 << ": " << x2 << ", " #x3 << ": " \
         << x3 << ", " #x4 << ": " << x4 << ", " #x5 << ": " << x5 << endl
#ifdef _DEBUG
#define debug(...)                                                        \
    CHOOSE((__VA_ARGS__, debug_5, debug_4, debug_3, debug_2, debug_1, ~)) \
    (__VA_ARGS__)
#else
#define debug(...)
#endif

void out() {
    cout << endl;
}
template <class T>
void out(const T &a) {
    cout << a;
    cout << endl;
}
template <class T, class... Ts>
void out(const T &a, const Ts &...b) {
    cout << a;
    (cout << ... << (cout << ' ', b));
    cout << endl;
}
#define rout_1(x1) return out(x1)
#define rout_2(x1, x2) return out(x1, x2)
#define rout_3(x1, x2, x3) return out(x1, x2, x3)
#define rout_4(x1, x2, x3, x4) return out(x1, x2, x3, x4)
#define rout_5(x1, x2, x3, x4, x5) return out(x1, x2, x3, x4, x5)
#define rout(...)                                                    \
    CHOOSE((__VA_ARGS__, rout_5, rout_4, rout_3, rout_2, rout_1, ~)) \
    (__VA_ARGS__)

struct fast_ios {
    fast_ios() {
        cin.tie(nullptr);
        ios::sync_with_stdio(false);
        cout << fixed << setprecision(12);
    };
} fast_ios_;

struct Binomial {
    int p;
    int MAX;
    vector<long long> fac, finv, inv;

    // テーブルを作る前処理
    Binomial(int p_, int n = 1) : p(p_), MAX(1), fac(2), finv(2), inv(2) {
        fac[0] = fac[1] = 1;
        finv[0] = finv[1] = 1;
        inv[1] = 1;
        if(n != 1) build(n);
    }

    void build(int new_max) {
        MAX++;
        fac.resize(new_max + 1);
        inv.resize(new_max + 1);
        finv.resize(new_max + 1);
        for(; MAX <= new_max; MAX++) {
            fac[MAX] = fac[MAX - 1] * MAX % p;
            inv[MAX] = p - inv[p % MAX] * (p / MAX) % p;
            finv[MAX] = finv[MAX - 1] * inv[MAX] % p;
        }
        MAX--;
    }

    // nCk
    long long mod_comb(int n, int k) {
        if(n < k) return 0;
        if(n < 0 || k < 0) return 0;
        if(n > MAX) build(n);
        return fac[n] * (finv[k] * finv[n - k] % p) % p;
    }
    long long operator()(int n, int k) {
        return mod_comb(n, k);
    }

    // nPk
    long long mod_perm(int n, int k) {
        if(n < k) return 0;
        if(n < 0 || k < 0) return 0;
        if(n > MAX) build(n);
        return fac[n] * finv[n - k] % p;
    }
    // n!
    long long operator[](int n) {
        if(n > MAX) build(n);
        return fac[n];
    }
    // 1/n!
    long long operator()(int n) {
        if(n > MAX) build(n);
        return finv[n];
    }
};

template <typename T = long long>
struct modpow {
    long long x, m;
    int n;
    vector<T> d;
    modpow(long long x) : x(x), n(1), d(1, 1) {}
    T operator[](int i) {
        while(n <= i) d.push_back(d.back() * x), ++n;
        return d[i];
    }
};
modpow two(2), ten(10);

struct RandomNumberGenerator {
    mt19937 mt;

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

    long long operator()(long long a, long long b) {  // [a, b)
        uniform_int_distribution<long long> dist(a, b - 1);
        return dist(mt);
    }

    long long operator()(long long b) {  // [0, b)
        return (*this)(0, b);
    }

    long long operator()() {
        return (*this)(0, 1LL << 60);
    }

    double operator[](double a) {
        return (double)(*this)(0, 1LL << 60) / (1LL << 60) * a;
    }

    double normal_dist(double sigma, double mean = 0) {
        std::normal_distribution<> dist(mean, sigma);
        return dist(mt);
    }
} rnd;

clock_t start_time = clock();
double now_time() {
    clock_t end_time = clock();
    return (double)(end_time - start_time) / CLOCKS_PER_SEC;
}

void input_graph(vector<vector<int>> &g, int m = -1, int bidirected = true) {
    if(m == -1) m = g.size() - 1;
    for(int i = 0; i < m; i++) {
        int u, v;
        cin >> u >> v;
        u--;
        v--;
        g[u].push_back(v);
        if(bidirected) g[v].push_back(u);
    }
}
vector<int> iota(int n, int s = 0) {
    vi a(n);
    iota(a.begin(), a.end(), s);
    return a;
}
template <class T>
void sort(vector<T> &v) {
    sort(all(v));
}
template <class T>
void rsort(vector<T> &v) {
    sort(rall(v));
}
template <class T>
void reverse(T &v) {
    reverse(all(v));
}
template <class T>
auto max(const T &a) {
    return *max_element(a.begin(), a.end());
}
template <class T>
auto min(const T &a) {
    return *min_element(a.begin(), a.end());
}
template <class T>
int max_id(const T &a) {
    return max_element(a.begin(), a.end()) - a.begin();
}
template <class T>
int min_id(const T &a) {
    return min_element(a.begin(), a.end()) - a.begin();
}
long long max(signed x, long long y) {
    return max((long long)x, y);
}
long long max(long long x, signed y) {
    return max(x, (long long)y);
}
long long min(signed x, long long y) {
    return min((long long)x, y);
}
long long min(long long x, signed y) {
    return min(x, (long long)y);
}
template <class T, class S>
bool chmax(T &a, const S &b) {
    if(a < (T)b) {
        a = (T)b;
        return 1;
    }
    return 0;
}
template <class T, class S>
bool chmin(T &a, const S &b) {
    if((T)b < a) {
        a = (T)b;
        return 1;
    }
    return 0;
}
template <class T>
vector<T> uniq(vector<T> v) {
    sort(v.begin(), v.end());
    v.erase(unique(v.begin(), v.end()), v.end());
    return v;
}
template <class T>
vector<T> compress(vector<T> v) {
    vector<T> v2(v.size());
    v2 = v;
    sort(v.begin(), v.end());
    v.erase(unique(v.begin(), v.end()), v.end());

    for(int i = 0; i < (int)v2.size(); i++) {
        v2[i] = lower_bound(v.begin(), v.end(), v2[i]) - v.begin();
    }
    return v2;
}
vector<int> inverse(vector<int> &p) {
    int n = p.size();
    vector<int> inv(n);
    for(int i = 0; i < n; i++) inv[p[i]] = i;
    return inv;
}
template <typename T>
vector<pair<T, int>> idx_pair(vector<T> &a) {
    int n = a.size();
    vector<pair<T, int>> res(n);
    for(int i = 0; i < n; i++) res[i] = {a[i], i};
    return res;
}
template <typename T>
vector<T> acc0(vector<T> &v) {
    vector<T> res(v.size());
    if((int)v.size() == 0) return res;
    res[0] = v[0];
    for(int i = 1; i < (int)v.size(); i++) {
        res[i] = res[i - 1] + v[i];
    }
    return res;
}
template <typename T>
vector<T> acc1(vector<T> &v) {
    vector<T> res(v.size() + 1);
    for(int i = 0; i < (int)v.size(); i++) {
        res[i + 1] = res[i] + v[i];
    }
    return res;
}
template <typename T>
vector<vector<T>> acc0(vector<vector<T>> v) {
    int h = v.size(), w = v[0].size();
    for(int i = 0; i < h; i++) {
        for(int j = 1; j < w; j++) {
            v[i][j] += v[i][j - 1];
        }
    }
    for(int i = 1; i < h; i++) {
        for(int j = 0; j < w; j++) {
            v[i][j] += v[i - 1][j];
        }
    }
    return v;
}
template <typename T>
vector<vector<T>> acc1(vector<vector<T>> &v) {
    int h = v.size(), w = v[0].size();
    vector<vector<T>> res(h + 1, vector<T>(w + 1));
    for(int i = 0; i < h; i++) {
        for(int j = 0; j < w; j++) {
            res[i + 1][j + 1] = v[i][j] + res[i + 1][j];
        }
    }
    for(int i = 0; i < h; i++) {
        for(int j = 0; j < w; j++) {
            res[i + 1][j + 1] += res[i][j + 1];
        }
    }
    return res;
}
template <class T>
void erase1(multiset<T> &st, int x) {
    auto it = st.find(x);
    assert(it != st.end());
    st.erase(it);
}

long long exp(long long x, int n) {
    long long res = 1;
    while(n > 0) {
        if(n & 1) res = res * x;
        x = x * x;
        n >>= 1;
    }
    return res;
}
int countDigits(long long n) {
    string tmp = to_string(n);
    return (int)tmp.size();
}
long long sq(long long n) {
    return n * n;
}
long long ceil(long long x, long long y) {
    return (x + y - 1) / y;
}
long long floor(long long x, long long y) {
    return (y < 0 ? floor(-x, -y)
                  : (x > 0 ? x / y : x / y - (x % y == 0 ? 0 : 1)));
}
constexpr long long tri(long long n) {
    return n * (n + 1) / 2;
}
// l + ... + r
constexpr long long tri(long long l, long long r) {
    return (l + r) * (r - l + 1) / 2;
}
int ctoi(const char &c, const char start = '0') {
    return c - start;
}
int atoi(const char &c, const char start = 'a') {
    return c - start;
}
vector<int> ctoi(string &s, const char start = '0') {
    vector<int> res;
    for(auto &c : s) {
        int x = c - start;
        if(x < 0 || x >= 10) x = -1;
        res.push_back(x);
    }
    return res;
}
vector<int> atoi(string &s, const char start = 'a') {
    vector<int> res;
    for(auto &c : s) {
        int x = c - start;
        if(x < 0 || x >= 26) x = -1;
        res.push_back(x);
    }
    return res;
}
int mex(vector<int> &a) {
    int n = a.size();
    vector<int> cnt(n + 1);
    for(int i = 0; i < n; i++) {
        if(a[i] > n) continue;
        cnt[a[i]]++;
    }
    int res = 0;
    while(cnt[res]) res++;
    return res;
}
void yes() {
    cout << "Yes" << endl;
}
void no() {
    cout << "No" << endl;
}
void yesno(bool x) {
    if(x)
        yes();
    else
        no();
}
void err() {
    cout << -1 << endl;
}

int dx[] = {1, 0, -1, 0, 1, 1, -1, -1};
int dy[] = {0, 1, 0, -1, -1, 1, 1, -1};

long long inf = (1 << 30) + (1LL << 60) - 2;
double eps = 1e-9;

// long long mod = 67280421310721;
// using mint = static_modint<1000000009>;
// using mint = dynamic_modint<1000000009>;
// long long mod = 1000000007;
// using mint = modint1000000007;
long long mod = 998244353;
using mint = modint998244353;
typedef vector<mint> vm;
typedef vector<vector<mint>> vvm;
typedef vector<vector<vector<mint>>> vvvm;
// Binomial C(mod);
// modpow<mint> mtwo(2), mten(10);
////////////////////////////////////////////////////////////////////////////////////////////
/*
蟻本p256を参考に
https://ei1333.github.io/luzhiled/snippets/math/matrix.html
を参考に
https://atcoder.jp/contests/abc253/submissions/32166405
を参考に(det)

Matrix<T>(int h,int w,T val=0):縦h,横w,中身valで初期化
unit:単位行列
演算:+,+=,-,-=,*,*=,==,[][]
(*はMarix,vector<T>,Tに対し、それぞれ定義してある）
(*vector<T>は、転置した縦ベクトルをかけた縦ベクトルを返していることに注意)

T det():行列式
Matrix Inv():逆行列。存在しない場合はMatrix()が返る。
vector<T> gauss_jordan（vector<T> B):
    Ax=bとなるx。Aの逆行列が存在しない場合、空の配列を返す。
coutで出力可
*/

template <typename T>
struct Matrix {
    int h, w;
    vector<vector<T>> d;
    Matrix() : h(0), w(0) {}
    Matrix(int h, int w) : h(h), w(w), d(h, vector<T>(w)) {}
    Matrix(int n) : h(n), w(n), d(n, vector<T>(n)) {}
    Matrix &unit() {
        assert(h == w);
        rep(i, h) d[i][i] = 1;
        return *this;
    }
    const vector<T> &operator[](int i) const {
        return d[i];
    }
    vector<T> &operator[](int i) {
        return d[i];
    }

    Matrix operator+(const Matrix &a) const {
        assert(w == a.h);
        Matrix r(h, a.w);
        for(int i = 0; i < h; i++)
            for(int j = 0; j < w; j++) {
                r[i][j] = d[i][j] + a[i][j];
            }
        return r;
    }

    Matrix &operator+=(const Matrix &B) {
        for(int i = 0; i < h; i++) {
            for(int j = 0; j < w; j++) {
                (*this)[i][j] += B[i][j];
            }
        }
        return *this;
    }

    Matrix operator-(const Matrix &a) const {
        assert(w == a.h);
        Matrix r(h, a.w);
        for(int i = 0; i < h; i++)
            for(int j = 0; j < w; j++) {
                r[i][j] = d[i][j] - a[i][j];
            }
        return r;
    }

    Matrix &operator-=(const Matrix &B) {
        for(int i = 0; i < h; i++) {
            for(int j = 0; j < w; j++) {
                (*this)[i][j] -= B[i][j];
            }
        }
        return *this;
    }

    Matrix operator*(const Matrix &a) const {
        assert(w == a.h);
        Matrix r(h, a.w);

        for(int i = 0; i < h; i++) {
            for(int k = 0; k < w; k++) {
                for(int j = 0; j < a.w; j++) {
                    r[i][j] += d[i][k] * a[k][j];
                }
            }
        }
        return r;
    }

    Matrix &operator*=(const Matrix &a) {
        assert(w == a.h);
        vector<vector<T>> r(h, vector<T>(w));
        for(int i = 0; i < h; i++) {
            for(int k = 0; k < w; k++) {
                for(int j = 0; j < a.w; j++) {
                    r[i][j] += (*this)[i][k] * a[k][j];
                }
            }
        }
        d.swap(r);
        return (*this);
    }

    vector<T> operator*(const vector<T> &B) const {
        vector<T> C(h);
        assert((int)B.size() == w);
        for(int i = 0; i < h; i++) {
            for(int j = 0; j < w; j++) {
                C[i] += (*this)[i][j] * B[j];
            }
        }
        return C;
    }
    Matrix operator*(const T &x) const {
        Matrix C(h, w);
        for(int i = 0; i < h; i++) {
            for(int j = 0; j < w; j++) {
                C[i][j] += (*this)[i][j] * x;
            }
        }
        return C;
    }
    Matrix &operator*=(const T &x) {
        vector<vector<T>> r(h, vector<T>(w));
        for(int i = 0; i < h; i++)
            for(int j = 0; j < w; j++) {
                r[i][j] += (*this)[i][j] * x;
            }
        d.swap(r);
        return (*this);
    }

    bool operator==(const Matrix &b) {
        if(h != b.h) return false;
        if(w != b.w) return false;
        for(int i = 0; i < h; i++) {
            for(int j = 0; j < w; j++) {
                if((*this)[i][j] != b[i][j]) return false;
            }
        }
        return true;
    }

    T det() {
        assert(h == w);
        T res = 1;
        rep(k, h) {
            for(int i = k; i < h; ++i) {
                if(d[i][k] == 0) continue;
                if(i != k) {
                    swap(d[i], d[k]);
                    res = -res;
                }
            }
            if(d[k][k] == 0) return 0;
            res *= d[k][k];
            T inv = T(1) / d[k][k];
            for(int j = 0; j < h; j++) d[k][j] *= inv;
            for(int i = k + 1; i < h; ++i) {
                T c = d[i][k];
                for(int j = k; j < h; ++j) d[i][j] -= d[k][j] * c;
            }
        }
        return res;
    }

    Matrix pow(long long t) const {
        assert(h == w);
        if(!t) return Matrix(h, h).unit();
        if(t == 1) return *this;
        Matrix r = pow(t >> 1);
        r = r * r;
        if(t & 1) r = r * (*this);
        return r;
    }

    // 逆行列を返す。ガウスの消去法。存在しない場合は長さ0の配列を返す。
    Matrix Inv() {
        assert(h == w);
        int n = h;
        Matrix B(n, 2 * n);
        for(int i = 0; i < n; i++) {
            for(int j = 0; j < n; j++) {
                B[i][j] = d[i][j];
            }
            B[i][i + n] = 1;
        }

        for(int i = 0; i < n; i++) {
            // 注目している変数の係数の絶対値が大きい式をi番目に持ってくる
            int pivot = i;
            for(int j = i; j < n; j++) {
                if(B[j][i] != 0) {
                    pivot = j;
                    break;
                }
            }
            swap(B[i], B[pivot]);

            // 逆行列存在しない
            if(B[i][i] == 0) return Matrix();

            for(int j = i + 1; j < 2 * n; j++) B[i][j] /= B[i][i];
            for(int j = 0; j < n; j++) {
                if(i != j) {
                    for(int k = i + 1; k < 2 * n; k++)
                        B[j][k] -= B[j][i] * B[i][k];
                }
            }
        }
        Matrix res(n, n);
        for(int i = 0; i < n; i++) {
            for(int j = 0; j < n; j++) res[i][j] = B[i][j + n];
        }
        return res;
    }

    // Ax=bとなるx。Aの逆行列が存在しない場合、空の配列を返す。
    vector<T> gauss_jordan(vector<T> b) {
        Matrix inv = (*this).Inv();
        if(inv.h == 0) return {};
        b = inv * b;
        return b;
    }

    friend ostream &operator<<(ostream &os, Matrix &p) {
        for(int i = 0; i < p.h; i++) {
            for(int j = 0; j < p.w; j++) {
                os << p[i][j] << " ";
            }
            os << endl;
        }
        return (os);
    }
};
////////////////////////////////////////////////////////////////////////////////////////////

signed main() {
    int testcase = 1;
    cin >> testcase;
    for(int i = 0; i < testcase; i++) {
        solve();
    }
}

void solve() {
    int n, m;
    cin >> n >> m;
    // swap(n, m);

    Matrix<mint> mat(3);
    mat[0][0] = 1;
    mat[0][1] = 1;
    mat[0][2] = 1;

    mat[1][0] = n;
    mat[1][1] = n - 1;
    mat[1][2] = n - 2;

    mat[2][0] = n * (n - 3) / 2;
    mat[2][1] = (n - 3) * (n - 2) / 2;
    mat[2][2] = ((n >= 5) ? (n - 2) * (n - 5) / 2 : 0) +
                ((n < 3) ? 2 : ((n == 4) ? 1 : 2));

    vm x = {1, 0, 0};
    // debug(mat);
    auto res = mat.pow(m) * x;

    out(res[0] + res[1] + res[2]);
}