#include <bits/stdc++.h>
#define pb push_back
#define eb emplace_back
#define fi first
#define se second
#define rep(i,N) for(long long i = 0; i < (long long)(N); i++)
#define repr(i,N) for(long long i = (long long)(N) - 1; i >= 0; i--)
#define rep1(i,N) for(long long i = 1; i <= (long long)(N) ; i++)
#define repr1(i,N) for(long long i = (N) ; (long long)(i) > 0 ; i--)
#define each(x,v) for(auto& x : v)
#define all(v) (v).begin(),(v).end()
#define sz(v) ((int)(v).size())
#define ini(...) int __VA_ARGS__; in(__VA_ARGS__)
#define inl(...) long long __VA_ARGS__; in(__VA_ARGS__)
#define ins(...) string __VA_ARGS__; in(__VA_ARGS__)
using namespace std; void solve();
using ll = long long; template<class T = ll> using V = vector<T>;
using vi = V<int>; using vl = V<>; using vvi = V< V<int> >;
constexpr int inf = 1001001001; constexpr ll infLL = (1LL << 61) - 1;
struct IoSetupNya {IoSetupNya() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); cerr << fixed << setprecision(7);} } iosetupnya;
template<typename T, typename U> inline bool amin(T &x, U y) { return (y < x) ? (x = y, true) : false; }
template<typename T, typename U> inline bool amax(T &x, U y) { return (x < y) ? (x = y, true) : false; }
template<typename T, typename U> ostream& operator <<(ostream& os, const pair<T, U> &p) { os << p.first << " " << p.second; return os; }
template<typename T, typename U> istream& operator >>(istream& is, pair<T, U> &p) { is >> p.first >> p.second; return is; }
template<typename T> ostream& operator <<(ostream& os, const vector<T> &v) { int s = (int)v.size(); rep(i,s) os << (i ? " " : "") << v[i]; return os; }
template<typename T> istream& operator >>(istream& is, vector<T> &v) { for(auto &x : v) is >> x; return is; }
void in(){} template <typename T,class... U> void in(T &t,U &...u){ cin >> t; in(u...);}
void out(){cout << "\n";} template <typename T,class... U> void out(const T &t,const U &...u){ cout << t; if(sizeof...(u)) cout << " "; out(u...);}
template<typename T>void die(T x){out(x); exit(0);}
#ifdef NyaanDebug
  #include "NyaanDebug.h"
  #define trc(...) do { cerr << #__VA_ARGS__ << " = "; dbg_out(__VA_ARGS__);} while(0)
  #define trca(v,N) do { cerr << #v << " = "; array_out(v , N);cout << endl;} while(0)
#else
  #define trc(...)
  #define trca(...)
  int main(){solve();}
#endif

#define in2(N,s,t) rep(i,N){in(s[i] , t[i]);}
#define in3(N,s,t,u) rep(i,N){in(s[i] , t[i] , u[i]);}
using vd = V<double>; using vs = V<string>; using vvl = V< V<> >;
template<typename T>using heap = priority_queue< T , V<T> , greater<T> >;

using P = pair<int,int>; using vp = V<P>;
constexpr int MOD = /**/ 1000000007; //*/ 998244353;
//////////////////

template< int mod >
struct ModInt {
  int x;

  ModInt() : x(0) {}

  ModInt(int64_t y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}

  ModInt &operator+=(const ModInt &p) {
    if((x += p.x) >= mod) x -= mod;
    return *this;
  }

  ModInt &operator-=(const ModInt &p) {
    if((x += mod - p.x) >= mod) x -= mod;
    return *this;
  }

  ModInt &operator*=(const ModInt &p) {
    x = (int) (1LL * x * p.x % mod);
    return *this;
  }

  ModInt &operator/=(const ModInt &p) {
    *this *= p.inverse();
    return *this;
  }

  ModInt operator-() const { return ModInt(-x); }

  ModInt operator+(const ModInt &p) const { return ModInt(*this) += p; }

  ModInt operator-(const ModInt &p) const { return ModInt(*this) -= p; }

  ModInt operator*(const ModInt &p) const { return ModInt(*this) *= p; }

  ModInt operator/(const ModInt &p) const { return ModInt(*this) /= p; }

  bool operator==(const ModInt &p) const { return x == p.x; }

  bool operator!=(const ModInt &p) const { return x != p.x; }

  ModInt inverse() const {
    int a = x, b = mod, u = 1, v = 0, t;
    while(b > 0) {
      t = a / b;
      swap(a -= t * b, b);
      swap(u -= t * v, v);
    }
    return ModInt(u);
  }

  ModInt pow(int64_t n) const {
    ModInt ret(1), mul(x);
    while(n > 0) {
      if(n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }

  friend ostream &operator<<(ostream &os, const ModInt &p) {
    return os << p.x;
  }

  friend istream &operator>>(istream &is, ModInt &a) {
    int64_t t;
    is >> t;
    a = ModInt< mod >(t);
    return (is);
  }

  static int get_mod() { return mod; }
};

using modint = ModInt< MOD >;

namespace FastFourierTransform {
  using real = double;

  struct C {
    real x, y;

    C() : x(0), y(0) {}

    C(real x, real y) : x(x), y(y) {}

    inline C operator+(const C &c) const { return C(x + c.x, y + c.y); }

    inline C operator-(const C &c) const { return C(x - c.x, y - c.y); }

    inline C operator*(const C &c) const { return C(x * c.x - y * c.y, x * c.y + y * c.x); }

    inline C conj() const { return C(x, -y); }
  };

  const real PI = acosl(-1);
  int base = 1;
  vector< C > rts = { {0, 0},
                     {1, 0} };
  vector< int > rev = {0, 1};


  void ensure_base(int nbase) {
    if(nbase <= base) return;
    rev.resize(1 << nbase);
    rts.resize(1 << nbase);
    for(int i = 0; i < (1 << nbase); i++) {
      rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
    }
    while(base < nbase) {
      real angle = PI * 2.0 / (1 << (base + 1));
      for(int i = 1 << (base - 1); i < (1 << base); i++) {
        rts[i << 1] = rts[i];
        real angle_i = angle * (2 * i + 1 - (1 << base));
        rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i));
      }
      ++base;
    }
  }

  void fft(vector< C > &a, int n) {
    assert((n & (n - 1)) == 0);
    int zeros = __builtin_ctz(n);
    ensure_base(zeros);
    int shift = base - zeros;
    for(int i = 0; i < n; i++) {
      if(i < (rev[i] >> shift)) {
        swap(a[i], a[rev[i] >> shift]);
      }
    }
    for(int k = 1; k < n; k <<= 1) {
      for(int i = 0; i < n; i += 2 * k) {
        for(int j = 0; j < k; j++) {
          C z = a[i + j + k] * rts[j + k];
          a[i + j + k] = a[i + j] - z;
          a[i + j] = a[i + j] + z;
        }
      }
    }
  }

  vector< int64_t > multiply(const vector< int > &a, const vector< int > &b) {
    int need = (int) a.size() + (int) b.size() - 1;
    int nbase = 1;
    while((1 << nbase) < need) nbase++;
    ensure_base(nbase);
    int sz = 1 << nbase;
    vector< C > fa(sz);
    for(int i = 0; i < sz; i++) {
      int x = (i < (int) a.size() ? a[i] : 0);
      int y = (i < (int) b.size() ? b[i] : 0);
      fa[i] = C(x, y);
    }
    fft(fa, sz);
    C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0);
    for(int i = 0; i <= (sz >> 1); i++) {
      int j = (sz - i) & (sz - 1);
      C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r;
      fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r;
      fa[i] = z;
    }
    for(int i = 0; i < (sz >> 1); i++) {
      C A0 = (fa[i] + fa[i + (sz >> 1)]) * t;
      C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * rts[(sz >> 1) + i];
      fa[i] = A0 + A1 * s;
    }
    fft(fa, sz >> 1);
    vector< int64_t > ret(need);
    for(int i = 0; i < need; i++) {
      ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
    }
    return ret;
  }
};

template< typename T >
struct ArbitraryModConvolution {
  using real = FastFourierTransform::real;
  using C = FastFourierTransform::C;

  ArbitraryModConvolution() = default;

  vector< T > multiply(const vector< T > &a, const vector< T > &b, int need = -1) {
    if(need == -1) need = a.size() + b.size() - 1;
    int nbase = 0;
    while((1 << nbase) < need) nbase++;
    FastFourierTransform::ensure_base(nbase);
    int sz = 1 << nbase;
    vector< C > fa(sz);
    for(int i = 0; i < (int)a.size(); i++) {
      fa[i] = C(a[i].x & ((1 << 15) - 1), a[i].x >> 15);
    }
    fft(fa, sz);
    vector< C > fb(sz);
    if(a == b) {
      fb = fa;
    } else {
      for(int i = 0; i < (int)b.size(); i++) {
        fb[i] = C(b[i].x & ((1 << 15) - 1), b[i].x >> 15);
      }
      fft(fb, sz);
    }
    real ratio = 0.25 / sz;
    C r2(0, -1), r3(ratio, 0), r4(0, -ratio), r5(0, 1);
    for(int i = 0; i <= (sz >> 1); i++) {
      int j = (sz - i) & (sz - 1);
      C a1 = (fa[i] + fa[j].conj());
      C a2 = (fa[i] - fa[j].conj()) * r2;
      C b1 = (fb[i] + fb[j].conj()) * r3;
      C b2 = (fb[i] - fb[j].conj()) * r4;
      if(i != j) {
        C c1 = (fa[j] + fa[i].conj());
        C c2 = (fa[j] - fa[i].conj()) * r2;
        C d1 = (fb[j] + fb[i].conj()) * r3;
        C d2 = (fb[j] - fb[i].conj()) * r4;
        fa[i] = c1 * d1 + c2 * d2 * r5;
        fb[i] = c1 * d2 + c2 * d1;
      }
      fa[j] = a1 * b1 + a2 * b2 * r5;
      fb[j] = a1 * b2 + a2 * b1;
    }
    fft(fa, sz);
    fft(fb, sz);
    vector< T > ret(need);
    for(int i = 0; i < need; i++) {
      int64_t aa = llround(fa[i].x);
      int64_t bb = llround(fb[i].x);
      int64_t cc = llround(fa[i].y);
      aa = T(aa).x, bb = T(bb).x, cc = T(cc).x;
      ret[i] = aa + (bb << 15) + (cc << 30);
    }
    return ret;
  }
};

template< int mod >
struct NumberTheoreticTransform {

  int base, max_base, root;
  vector< int > rev, rts;

  NumberTheoreticTransform() : base(1), rev{0, 1}, rts{0, 1} {
    assert(mod >= 3 && mod % 2 == 1);
    auto tmp = mod - 1;
    max_base = 0;
    while(tmp % 2 == 0) tmp >>= 1, max_base++;
    root = 2;
    while(mod_pow(root, (mod - 1) >> 1) == 1) ++root;
    assert(mod_pow(root, mod - 1) == 1);
    root = mod_pow(root, (mod - 1) >> max_base);
  }

  inline int mod_pow(int x, int n) {
    int ret = 1;
    while(n > 0) {
      if(n & 1) ret = mul(ret, x);
      x = mul(x, x);
      n >>= 1;
    }
    return ret;
  }

  inline int inverse(int x) {
    return mod_pow(x, mod - 2);
  }

  inline unsigned add(unsigned x, unsigned y) {
    x += y;
    if(x >= mod) x -= mod;
    return x;
  }

  inline unsigned mul(unsigned a, unsigned b) {
    return 1ull * a * b % (unsigned long long) mod;
  }

  void ensure_base(int nbase) {
    if(nbase <= base) return;
    rev.resize(1 << nbase);
    rts.resize(1 << nbase);
    for(int i = 0; i < (1 << nbase); i++) {
      rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
    }
    assert(nbase <= max_base);
    while(base < nbase) {
      int z = mod_pow(root, 1 << (max_base - 1 - base));
      for(int i = 1 << (base - 1); i < (1 << base); i++) {
        rts[i << 1] = rts[i];
        rts[(i << 1) + 1] = mul(rts[i], z);
      }
      ++base;
    }
  }


  void ntt(vector< int > &a) {
    const int n = (int) a.size();
    assert((n & (n - 1)) == 0);
    int zeros = __builtin_ctz(n);
    ensure_base(zeros);
    int shift = base - zeros;
    for(int i = 0; i < n; i++) {
      if(i < (rev[i] >> shift)) {
        swap(a[i], a[rev[i] >> shift]);
      }
    }
    for(int k = 1; k < n; k <<= 1) {
      for(int i = 0; i < n; i += 2 * k) {
        for(int j = 0; j < k; j++) {
          int z = mul(a[i + j + k], rts[j + k]);
          a[i + j + k] = add(a[i + j], mod - z);
          a[i + j] = add(a[i + j], z);
        }
      }
    }
  }


  vector< int > multiply(vector< int > a, vector< int > b) {
    int need = a.size() + b.size() - 1;
    int nbase = 1;
    while((1 << nbase) < need) nbase++;
    ensure_base(nbase);
    int sz = 1 << nbase;
    a.resize(sz, 0);
    b.resize(sz, 0);
    ntt(a);
    ntt(b);
    int inv_sz = inverse(sz);
    for(int i = 0; i < sz; i++) {
      a[i] = mul(a[i], mul(b[i], inv_sz));
    }
    reverse(a.begin() + 1, a.end());
    ntt(a);
    a.resize(need);
    return a;
  }

  vector<modint> multiply_for_fps(const vector<modint> &a,const vector<modint> &b){
    vector<int> A(a.size()) , B(b.size());
    for(int i = 0;i < (int)a.size(); i++) A[i] = a[i].x;
    for(int i = 0;i < (int)b.size(); i++) B[i] = b[i].x;
    auto C = multiply( A , B );
    vector<modint> ret(C.size());
    for(int i = 0; i < (int)C.size() ;i++) ret[i].x = C[i];
    return ret;
  }
};


template< typename T >
struct FormalPowerSeries : vector< T > {
  using vector< T >::vector;
  using P = FormalPowerSeries;

  using MULT = function< P(P, P) >;

  static MULT &get_mult() {
    static MULT mult = nullptr;
    return mult;
  }

  static void set_fft(MULT f) {
    get_mult() = f;
  }

  void shrink() {
    while(this->size() && this->back() == T(0)) this->pop_back();
  }

  P operator+(const P &r) const { return P(*this) += r; }

  P operator+(const T &v) const { return P(*this) += v; }

  P operator-(const P &r) const { return P(*this) -= r; }

  P operator-(const T &v) const { return P(*this) -= v; }

  P operator*(const P &r) const { return P(*this) *= r; }

  P operator*(const T &v) const { return P(*this) *= v; }

  P operator/(const P &r) const { return P(*this) /= r; }

  P operator%(const P &r) const { return P(*this) %= r; }

  P &operator+=(const P &r) {
    if(r.size() > this->size()) this->resize(r.size());
    for(int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
    return *this;
  }

  P &operator+=(const T &r) {
    if(this->empty()) this->resize(1);
    (*this)[0] += r;
    return *this;
  }

  P &operator-=(const P &r) {
    if(r.size() > this->size()) this->resize(r.size());
    for(int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
    shrink();
    return *this;
  }

  P &operator-=(const T &r) {
    if(this->empty()) this->resize(1);
    (*this)[0] -= r;
    shrink();
    return *this;
  }

  P &operator*=(const T &v) {
    const int n = (int) this->size();
    for(int k = 0; k < n; k++) (*this)[k] *= v;
    return *this;
  }

  P &operator*=(const P &r) {
    if(this->empty() || r.empty()) {
      this->clear();
      return *this;
    }
    assert(get_mult() != nullptr);
    return *this = get_mult()(*this, r);
  }

  P &operator%=(const P &r) {
    return *this -= *this / r * r;
  }

  P operator-() const {
    P ret(this->size());
    for(int i = 0; i < this->size(); i++) ret[i] = -(*this)[i];
    return ret;
  }

  P &operator/=(const P &r) {
    if(this->size() < r.size()) {
      this->clear();
      return *this;
    }
    int n = this->size() - r.size() + 1;
    return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n);
  }

  P pre(int sz) const {
    return P(begin(*this), begin(*this) + min((int) this->size(), sz));
  }

  P operator>>(int sz) const {
    if(this->size() <= sz) return {};
    P ret(*this);
    ret.erase(ret.begin(), ret.begin() + sz);
    return ret;
  }

  P operator<<(int sz) const {
    P ret(*this);
    ret.insert(ret.begin(), sz, T(0));
    return ret;
  }

  P rev(int deg = -1) const {
    P ret(*this);
    if(deg != -1) ret.resize(deg, T(0));
    reverse(begin(ret), end(ret));
    return ret;
  }

  P diff() const {
    const int n = (int) this->size();
    P ret(max(0, n - 1));
    for(int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * T(i);
    return ret;
  }

  P integral() const {
    const int n = (int) this->size();
    P ret(n + 1);
    ret[0] = T(0);
    for(int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1);
    return ret;
  }

  // F(0) must not be 0
  P inv(int deg = -1) const {
    assert(((*this)[0]) != T(0));
    const int n = (int) this->size();
    if(deg == -1) deg = n;
    P ret({T(1) / (*this)[0]});
    for(int i = 1; i < deg; i <<= 1) {
      ret = (ret + ret - ret * ret * pre(i << 1)).pre(i << 1);
    }
    return ret.pre(deg);
  }

  // F(0) must be 1
  P log(int deg = -1) const {
    assert((*this)[0] == 1);
    const int n = (int) this->size();
    if(deg == -1) deg = n;
    return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
  }

  P sqrt(int deg = -1) const {
    const int n = (int) this->size();
    if(deg == -1) deg = n;

    if((*this)[0] == T(0)) {
      for(int i = 1; i < n; i++) {
        if((*this)[i] != T(0)) {
          if(i & 1) return {};
          if(deg - i / 2 <= 0) break;
          auto ret = (*this >> i).sqrt(deg - i / 2) << (i / 2);
          if(ret.size() < deg) ret.resize(deg, T(0));
          return ret;
        }
      }
      return P(deg, 0);
    }

    P ret({T(1)});
    T inv2 = T(1) / T(2);
    for(int i = 1; i < deg; i <<= 1) {
      ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2;
    }
    return ret.pre(deg);
  }

  // F(0) must be 0
  P exp(int deg = -1) const {
    assert((*this)[0] == T(0));
    const int n = (int) this->size();
    if(deg == -1) deg = n;
    P ret({T(1)});
    for(int i = 1; i < deg; i <<= 1) {
      ret = (ret * (pre(i << 1) + T(1) - ret.log(i << 1))).pre(i << 1);
    }
    return ret.pre(deg);
  }

  P pow(int64_t k, int deg = -1) const {
    const int n = (int) this->size();
    if(deg == -1) deg = n;
    for(int i = 0; i < n; i++) {
      if((*this)[i] != T(0)) {
        T rev = T(1) / (*this)[i];
        P C(*this * rev);
        P D(n - i);
        for(int j = i; j < n; j++) D[j - i] = C[j];
        D = (D.log() * k).exp() * (*this)[i].pow(k);
        P E(deg);
        if(i * k > deg) return E;
        auto S = i * k;
        for(int j = 0; j + S < deg && j < D.size(); j++) E[j + S] = D[j];
        return E;
      }
    }
    return *this;
  }

  T eval(T x) const {
    T r = 0, w = 1;
    for(auto &v : *this) {
      r += w * v;
      w *= x;
    }
    return r;
  }
};

using FPS = FormalPowerSeries< modint >;

// fにa * x^n + bを掛ける
void mul_simple(FPS &f,modint a ,int n, modint b){
  for(int i = (int)f.size() - 1 ; i >= 0 ; i--){
    f[i] *= b;
    if(i >= n) f[i] += f[i - n] * a; 
  }
}

// fからa * x^n + bを割る
void div_simple(FPS &f,modint a,int n,modint b){
  for(int i = 0 ; i < (int)f.size() ; i++){
    f[i] /= b;
    if(i + n < (int)f.size() ) f[n + i] -= f[i] * a; 
  }
}

// f / gをdeg(f)次まで求める
FPS div_(FPS &f , FPS g){
  int n = f.size();
  return (f * g.inv(n)).pre(n);
}

// solve関数内で
//
// FPS::set_fft(mul);
//
// とすること。


// 下記のリンクを実装(kitamasa法のモンゴメリ乗算を使わない版)
// http://q.c.titech.ac.jp/docs/progs/polynomial_division.html
// k項間漸化式のa_Nを求める O(k log k log N)
// N ... 求めたい項　(0-indexed)
// Q ... 漸化式 (1 - \sum_i c_i x^i)の形
// a ... 初期解 (a_0 , a_1 , ... , a_k-1)
// x^N を fでわった剰余を求め、aと内積を取る
modint kitamasa(ll N, FPS &Q, FPS &a){
  int k = Q.size() - 1;
  assert( (int)a.size() == k );
  FPS P = a * Q; P.resize(k);
  while(N){
    auto Q2 = Q;
    for(int i = 1; i < (int)Q2.size(); i += 2) Q2[i].x = MOD - Q2[i].x;
    auto S = P * Q2;
    auto T = Q * Q2;
    if(N & 1){
      for(int i = 1 ; i < (int)S.size() ; i += 2) P[i>>1].x = S[i].x;
      for(int i = 0 ; i < (int)T.size() ; i += 2) Q[i>>1].x = T[i].x;
    }
    else{
      for(int i = 0 ; i < (int)S.size() ; i += 2) P[i>>1].x = S[i].x;
      for(int i = 0 ; i < (int)T.size() ; i += 2) Q[i>>1].x = T[i].x;
    }
    N >>= 1;
  }
  return P[0];
}


// 素数判定 O( sqrt(N) log log N )
// 0からNに対して素数->1、それ以外->0の配列を返す関数
vector<int> Primes(int N){
  vector<int> A(N + 1 , 1);
  A[0] = A[1] = 0;
  for(int i = 2; i * i <= N ; i++) 
    if(A[i]==1) for(int j = i << 1 ; j <= N; j += i) A[j] = 0;
  return A;
}

// 因数 O( sqrt(N) log log N )
// 0からNに対して素数->1、それ以外->最小の素数である因数、の配列を返す
vector<int> Factors(int N){
  vector<int> A(N + 1 , 1);
  A[0] = A[1] = 0;
  for(int i = 2; i * i <= N ; i++) 
    if(A[i]==1) for(int j = i << 1 ; j <= N; j += i) A[j] = i;
  return A;
}

// オイラーのトーシェント関数 φ(N)=(Nと互いに素なN以下の自然数の個数)
vector<int> EulersTotientFunction(int N){
  vector<int> ret(N + 1 , 0);
  for(int i = 0; i <= N ; i++) ret[i] = i; 
  for(int i = 2 ; i <= N ; i++){
    if(ret[i] == i)
      for(int j = i; j <= N; j += i) ret[j] = ret[j] / i * (i - 1);
  }
  return ret;
}

// 約数列挙 O(sqrt(N))
// Nの約数を列挙した配列を返す
vector<long long> Divisor(long long N){
  vector<long long> v;
  for(long long i = 1; i * i <= N ; i++){
    if(N % i == 0){
      v.push_back(i);
      if(i * i != N) v.push_back(N / i);
    }
  }
  return v;
}

// 素因数分解
// 因数をkey、そのべきをvalueとするmapを返す
// ex) N=12 -> m={ (2,2) , (3,1) }
map<long long,int> PrimeFactors(long long N){
  map<long long,int> m;
  for(long long i=2; i * i <= N; i++)
    while(N % i == 0) m[i]++ , N /= i;
  if(N != 1) m[N]++;
  return m;
}

// 原始根 modでrが原始根かどうかを調べる
bool PrimitiveRoot(long long r , long long mod){
  r %= mod; if(r == 0) return false;
  auto modpow = [](long long a,long long b,long long m)->long long{
    a %= m; long long ret = 1;
    while(b){
      if(b & 1) ret = a * ret % m;
      a = a * a % m;
      b >>= 1;
    }
    return ret;
  };
  map<long long,int> m = PrimeFactors(mod - 1);
  each(x , m){
    if(modpow(r , (mod - 1) / x.fi , mod ) == 1) return false;
  }
  return true;
}

// 拡張ユークリッド ax+by=gcd(a,b)の解
// 返り値　最大公約数
long long extgcd(long long a,long long b, long long &x, long long &y){
  if(b == 0){
    x = 1; y = 0; return a;
  }
  long long d = extgcd(b , a%b , y , x);
  y -= a / b * x;
  return d;
}

// ブール代数ライブラリ
// Point. 乗法の単位元は-1 (UNIT & a = aを満たすUNITであるため)
struct BA{
  unsigned long long x;
  BA(): x(0){}
  BA(unsigned long long y):x(y){}
  
  BA operator += (const BA &p){
    x = x ^ p.x;
    return (*this);
  }
  BA operator *= (const BA &p){
    x = x & p.x;
    return (*this);
  }

  BA operator+(const BA &p)const {return BA(*this) += p;}
  BA operator*(const BA &p)const {return BA(*this) *= p;}
  bool operator==(const BA &p) const { return x == p.x; }
  bool operator!=(const BA &p) const { return x != p.x; }

  friend ostream &operator<<(ostream &os,const BA &p){
    return os << p.x;
  }
  friend istream &operator>>(istream &is, BA &a){
    unsigned int t;
    is >> t;
    a = BA(t);
    return (is);
  }
};

// c++17での名前衝突を避けるためdefine
#define gcd nyagcd
#define lcm nyalcm
ll nyagcd(ll x, ll y){
  ll z;
  if(x > y) swap(x,y);
  while(x){
    x = y % (z = x); y = z;
  }    
  return y;
}
ll nyalcm(ll x,ll y){
  return 1LL * x / gcd(x,y) * y;
}

vector<ll> fac,finv,inv;
void cominit(int MAX) {
  MAX++;
  fac.resize(MAX , 0);
  finv.resize(MAX , 0);
  inv.resize(MAX , 0);
  fac[0] = fac[1] = finv[0] = finv[1] = inv[1] = 1;
  for (int i = 2; i < MAX; i++){
    fac[i] = fac[i - 1] * i % MOD;
    inv[i] = MOD - inv[MOD%i] * (MOD / i) % MOD;
    finv[i] = finv[i - 1] * inv[i] % MOD;
  }
}
// nCk combination 
inline long long COM(int n,int k){
  if(n < k || k < 0 || n < 0) return 0;
  else return fac[n] * (finv[k] * finv[n - k] % MOD) % MOD;
}
// nPk permutation
inline long long PER(int n,int k){
  if (n < k || k < 0 || n < 0) return 0;
  else return (fac[n] * finv[n - k]) % MOD;
}
// nHk homogeneous polynomial
inline long long HGP(int n,int k){
  if(n == 0 && k == 0) return 1; //問題依存?
  else if(n < 1 || k < 0) return 0;
  else return fac[n + k - 1] * (finv[k] * finv[n - 1] % MOD) % MOD;
}

void solve(){
  ini(N);
  cominit(10000);
  modint ans = 0;
  vvi a(2048 , vi(2048 , 0));
  rep(i , N){
    ini(x, y);
    a[x][y]++;
    ans -= COM(2 * x + 2 * y , 2 * x);
  } 
  
  constexpr int D = 2000;
  FPS f(2 * D + 5);
  for(int k = D * 2 ; k > 0 ; k--){
    FPS g(2 * D + 5);
    for(int i = 1 ; i <= D ; i++){
      if(k - i < 0 || k - i > D) continue;
      if(a[i][k - i] != 0)
        g[D - i] += a[i][k - i];
    }
    f += g;
    mul_simple(f , 1 , 1 , 1);
  }
  FPS db = f; db.resize(10); trc(db);
  for(int i = 0 ; i <= 2 * D ; i++){
    ans += f[i] * f[2 * D - i];
  }
  trc(ans);
  ans /= 2;
  out(ans);


}

Measured memory_bytes: 17024