#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")
#pragma GCC optimize("inline")
#include<bits/stdc++.h>
using namespace std;
#define MD (1000000007U)
void*wmem;
char memarr[314572800];
template<class T> inline void walloc1d(T **arr, int x, void **mem = &wmem){
  static int skip[16] = {0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1};
  (*mem) = (void*)( ((char*)(*mem)) + skip[((unsigned long long)(*mem)) & 15] );
  (*arr)=(T*)(*mem);
  (*mem)=((*arr)+x);
}
template<class T> inline void walloc1d(T **arr, int x1, int x2, void **mem = &wmem){
  walloc1d(arr, x2-x1, mem);
  (*arr) -= x1;
}
template<class T> void malloc1d(T **arr, int x){
  (*arr) = (T*)malloc(x*sizeof(T));
}
template<class T> void free1d(T *arr){
  free(arr);
}
template<class T1> void sortA_L(int N, T1 a[], void *mem = wmem){
  sort(a, a+N);
}
template<class T1, class T2> void sortA_L(int N, T1 a[], T2 b[], void *mem = wmem){
  int i;
  pair<T1, T2>*arr;
  walloc1d(&arr, N, &mem);
  for(i=(0);i<(N);i++){
    arr[i].first = a[i];
    arr[i].second = b[i];
  }
  sort(arr, arr+N);
  for(i=(0);i<(N);i++){
    a[i] = arr[i].first;
    b[i] = arr[i].second;
  }
}
template<class T1, class T2, class T3> void sortA_L(int N, T1 a[], T2 b[], T3 c[], void *mem = wmem){
  int i;
  pair<T1, pair<T2, T3> >*arr;
  walloc1d(&arr, N, &mem);
  for(i=(0);i<(N);i++){
    arr[i].first = a[i];
    arr[i].second.first = b[i];
    arr[i].second.second = c[i];
  }
  sort(arr, arr+N);
  for(i=(0);i<(N);i++){
    a[i] = arr[i].first;
    b[i] = arr[i].second.first;
    c[i] = arr[i].second.second;
  }
}
struct Modint{
  unsigned val;
  Modint(){
    val=0;
  }
  Modint(int a){
    val = ord(a);
  }
  Modint(unsigned a){
    val = ord(a);
  }
  Modint(long long a){
    val = ord(a);
  }
  Modint(unsigned long long a){
    val = ord(a);
  }
  inline unsigned ord(unsigned a){
    return a%MD;
  }
  inline unsigned ord(int a){
    a %= (int)MD;
    if(a < 0){
      a += MD;
    }
    return a;
  }
  inline unsigned ord(unsigned long long a){
    return a%MD;
  }
  inline unsigned ord(long long a){
    a %= (int)MD;
    if(a < 0){
      a += MD;
    }
    return a;
  }
  inline unsigned get(){
    return val;
  }
  inline Modint &operator++(){
    val++;
    if(val >= MD){
      val -= MD;
    }
    return *this;
  }
  inline Modint &operator--(){
    if(val == 0){
      val = MD - 1;
    }
    else{
      --val;
    }
    return *this;
  }
  inline Modint operator++(int a){
    Modint res(*this);
    val++;
    if(val >= MD){
      val -= MD;
    }
    return res;
  }
  inline Modint operator--(int a){
    Modint res(*this);
    if(val == 0){
      val = MD - 1;
    }
    else{
      --val;
    }
    return res;
  }
  inline Modint &operator+=(Modint a){
    val += a.val;
    if(val >= MD){
      val -= MD;
    }
    return *this;
  }
  inline Modint &operator-=(Modint a){
    if(val < a.val){
      val = val + MD - a.val;
    }
    else{
      val -= a.val;
    }
    return *this;
  }
  inline Modint &operator*=(Modint a){
    val = ((unsigned long long)val*a.val)%MD;
    return *this;
  }
  inline Modint &operator/=(Modint a){
    return *this *= a.inverse();
  }
  inline Modint operator+(Modint a){
    return Modint(*this)+=a;
  }
  inline Modint operator-(Modint a){
    return Modint(*this)-=a;
  }
  inline Modint operator*(Modint a){
    return Modint(*this)*=a;
  }
  inline Modint operator/(Modint a){
    return Modint(*this)/=a;
  }
  inline Modint operator+(int a){
    return Modint(*this)+=Modint(a);
  }
  inline Modint operator-(int a){
    return Modint(*this)-=Modint(a);
  }
  inline Modint operator*(int a){
    return Modint(*this)*=Modint(a);
  }
  inline Modint operator/(int a){
    return Modint(*this)/=Modint(a);
  }
  inline Modint operator+(long long a){
    return Modint(*this)+=Modint(a);
  }
  inline Modint operator-(long long a){
    return Modint(*this)-=Modint(a);
  }
  inline Modint operator*(long long a){
    return Modint(*this)*=Modint(a);
  }
  inline Modint operator/(long long a){
    return Modint(*this)/=Modint(a);
  }
  inline Modint operator-(void){
    Modint res;
    if(val){
      res.val=MD-val;
    }
    else{
      res.val=0;
    }
    return res;
  }
  inline operator bool(void){
    return val!=0;
  }
  inline operator int(void){
    return get();
  }
  inline operator long long(void){
    return get();
  }
  inline Modint inverse(){
    int a = val;
    int b = MD;
    int u = 1;
    int v = 0;
    int t;
    Modint res;
    while(b){
      t = a / b;
      a -= t * b;
      swap(a, b);
      u -= t * v;
      swap(u, v);
    }
    if(u < 0){
      u += MD;
    }
    res.val = u;
    return res;
  }
  inline Modint pw(unsigned long long b){
    Modint a(*this);
    Modint res;
    res.val = 1;
    while(b){
      if(b&1){
        res *= a;
      }
      b >>= 1;
      a *= a;
    }
    return res;
  }
  inline bool operator==(int a){
    return ord(a)==val;
  }
  inline bool operator!=(int a){
    return ord(a)!=val;
  }
}
;
inline Modint operator+(int a, Modint b){
  return Modint(a)+=b;
}
inline Modint operator-(int a, Modint b){
  return Modint(a)-=b;
}
inline Modint operator*(int a, Modint b){
  return Modint(a)*=b;
}
inline Modint operator/(int a, Modint b){
  return Modint(a)/=b;
}
inline Modint operator+(long long a, Modint b){
  return Modint(a)+=b;
}
inline Modint operator-(long long a, Modint b){
  return Modint(a)-=b;
}
inline Modint operator*(long long a, Modint b){
  return Modint(a)*=b;
}
inline Modint operator/(long long a, Modint b){
  return Modint(a)/=b;
}
inline int my_getchar_unlocked(){
  static char buf[1048576];
  static int s = 1048576;
  static int e = 1048576;
  if(s == e && e == 1048576){
    e = fread_unlocked(buf, 1, 1048576, stdin);
    s = 0;
  }
  if(s == e){
    return EOF;
  }
  return buf[s++];
}
struct MY_WRITER{
  char buf[1048576];
  int s;
  int e;
  MY_WRITER(){
    s = 0;
    e = 1048576;
  }
  ~MY_WRITER(){
    if(s){
      fwrite_unlocked(buf, 1, s, stdout);
    }
  }
}
;
MY_WRITER MY_WRITER_VAR;
void my_putchar_unlocked(int a){
  if(MY_WRITER_VAR.s == MY_WRITER_VAR.e){
    fwrite_unlocked(MY_WRITER_VAR.buf, 1, MY_WRITER_VAR.s, stdout);
    MY_WRITER_VAR.s = 0;
  }
  MY_WRITER_VAR.buf[MY_WRITER_VAR.s++] = a;
}
inline void wt_L(char a){
  my_putchar_unlocked(a);
}
inline void wt_L(int x){
  int s=0;
  int m=0;
  char f[10];
  if(x<0){
    m=1;
    x=-x;
  }
  while(x){
    f[s++]=x%10;
    x/=10;
  }
  if(!s){
    f[s++]=0;
  }
  if(m){
    my_putchar_unlocked('-');
  }
  while(s--){
    my_putchar_unlocked(f[s]+'0');
  }
}
inline void wt_L(unsigned x){
  int s=0;
  char f[10];
  while(x){
    f[s++]=x%10;
    x/=10;
  }
  if(!s){
    f[s++]=0;
  }
  while(s--){
    my_putchar_unlocked(f[s]+'0');
  }
}
inline void wt_L(long long x){
  int s=0;
  int m=0;
  char f[20];
  if(x<0){
    m=1;
    x=-x;
  }
  while(x){
    f[s++]=x%10;
    x/=10;
  }
  if(!s){
    f[s++]=0;
  }
  if(m){
    my_putchar_unlocked('-');
  }
  while(s--){
    my_putchar_unlocked(f[s]+'0');
  }
}
inline void wt_L(unsigned long long x){
  int s=0;
  char f[21];
  while(x){
    f[s++]=x%10;
    x/=10;
  }
  if(!s){
    f[s++]=0;
  }
  while(s--){
    my_putchar_unlocked(f[s]+'0');
  }
}
inline void wt_L(Modint x){
  int i;
  i = (int)x;
  wt_L(i);
}
int WRITER_DOUBLE_DIGIT = 15;
inline int writerDigit_double(){
  return WRITER_DOUBLE_DIGIT;
}
inline void writerDigit_double(int d){
  WRITER_DOUBLE_DIGIT = d;
}
inline void wt_L(double x){
  const int d = WRITER_DOUBLE_DIGIT;
  int k;
  int r;
  double v;
  if(x!=x || (x==x+1 && x==2*x)){
    my_putchar_unlocked('E');
    my_putchar_unlocked('r');
    my_putchar_unlocked('r');
    return;
  }
  if(x < 0){
    my_putchar_unlocked('-');
    x = -x;
  }
  x += 0.5 * pow(0.1, d);
  r = 0;
  v = 1;
  while(x >= 10*v){
    v *= 10;
    r++;
  }
  while(r >= 0){
    r--;
    k = floor(x / v);
    if(k >= 10){
      k = 9;
    }
    if(k <= -1){
      k = 0;
    }
    x -= k * v;
    v *= 0.1;
    my_putchar_unlocked(k + '0');
  }
  if(d > 0){
    my_putchar_unlocked('.');
    v = 1;
    for(r=(0);r<(d);r++){
      v *= 0.1;
      k = floor(x / v);
      if(k >= 10){
        k = 9;
      }
      if(k <= -1){
        k = 0;
      }
      x -= k * v;
      my_putchar_unlocked(k + '0');
    }
  }
}
inline void wt_L(const char c[]){
  int i=0;
  for(i=0;c[i]!='\0';i++){
    my_putchar_unlocked(c[i]);
  }
}
inline void wt_L(string &x){
  int i=0;
  for(i=0;x[i]!='\0';i++){
    my_putchar_unlocked(x[i]);
  }
}
template<class S, class T> inline S chmax(S &a, T b){
  if(a<b){
    a=b;
  }
  return a;
}
template<class T> struct Comb{
  int mem_fact;
  T*factri;
  T*ifactri;
  int mem_dfact;
  T*dfactri;
  int mem_pw2;
  int mem_pw3;
  int mem_pw10;
  int mem_rep1;
  T*pw2c;
  T*pw3c;
  T*pw10c;
  T*rep1c;
  int mem_ipw2;
  int mem_ipw3;
  int mem_ipw10;
  T*ipw2c;
  T*ipw3c;
  T*ipw10c;
  Comb(){
    mem_fact = 0;
    mem_dfact = 0;
    mem_pw2 = mem_pw3 = mem_pw10 = mem_rep1 = 0;
    mem_ipw2 = mem_ipw3 = mem_ipw10 = 0;
  }
  inline void expand_fact(int k){
    int i;
    if(k <= mem_fact){
      return;
    }
    chmax(k, 2 * mem_fact);
    if(mem_fact == 0){
      factri = (T*)malloc(k * sizeof(T));
      ifactri = (T*)malloc(k * sizeof(T));
      factri[0] = 1;
      for(i=(1);i<(k);i++){
        factri[i] = i * factri[i-1];
      }
      ifactri[k-1] = 1 / factri[k-1];
      for(i=(k-1)-1;i>=(0);i--){
        ifactri[i] = (i+1) * ifactri[i+1];
      }
    }
    else{
      factri = (T*)realloc(factri, k * sizeof(T));
      ifactri = (T*)realloc(ifactri, k * sizeof(T));
      for(i=(mem_fact);i<(k);i++){
        factri[i] = i * factri[i-1];
      }
      ifactri[k-1] = 1 / factri[k-1];
      for(i=(k-1)-1;i>=(mem_fact);i--){
        ifactri[i] = (i+1) * ifactri[i+1];
      }
    }
    mem_fact = k;
  }
  inline T fac(int k){
    if(mem_fact < k+1){
      expand_fact(k+1);
    }
    return factri[k];
  }
  inline T ifac(int k){
    if(mem_fact < k+1){
      expand_fact(k+1);
    }
    return ifactri[k];
  }
  inline T C(int a, int b){
    if(b < 0 || b > a){
      return 0;
    }
    if(mem_fact < a+1){
      expand_fact(a+1);
    }
    return factri[a] * ifactri[b] * ifactri[a-b];
  }
  inline T P(int a, int b){
    if(b < 0 || b > a){
      return 0;
    }
    if(mem_fact < a+1){
      expand_fact(a+1);
    }
    return factri[a] * ifactri[a-b];
  }
  inline T H(int a, int b){
    if(a==0 && b==0){
      return 1;
    }
    if(a <= 0 || b < 0){
      return 0;
    }
    if(mem_fact < a+b){
      expand_fact(a+b);
    }
    return C(a+b-1, b);
  }
  inline T Multinomial(int sz, int a[]){
    int i;
    int s = 0;
    T res;
    for(i=(0);i<(sz);i++){
      s += a[i];
    }
    if(mem_fact < s+1){
      expand_fact(s+1);
    }
    res = factri[s];
    for(i=(0);i<(sz);i++){
      res *= ifactri[a[i]];
    }
    return res;
  }
  inline T Multinomial(int a){
    return 1;
  }
  inline T Multinomial(int a, int b){
    if(mem_fact < a+b+1){
      expand_fact(a+b+1);
    }
    return factri[a+b] * ifactri[a] * ifactri[b];
  }
  inline T Multinomial(int a, int b, int c){
    if(mem_fact < a+b+c+1){
      expand_fact(a+b+c+1);
    }
    return factri[a+b+c] * ifactri[a] * ifactri[b] * ifactri[c];
  }
  inline T Multinomial(int a, int b, int c, int d){
    if(mem_fact < a+b+c+d+1){
      expand_fact(a+b+c+d+1);
    }
    return factri[a+b+c+d] * ifactri[a] * ifactri[b] * ifactri[c] * ifactri[d];
  }
  inline T Catalan(int n){
    if(n < 0){
      return 0;
    }
    if(mem_fact < 2*n+1){
      expand_fact(2*n+1);
    }
    return factri[2*n] * ifactri[n] * ifactri[n+1];
  }
  inline T Catalan(int n, int m, int k){
    if(k <= 0){
      return C(n+m, n);
    }
    if(n < k || m < k){
      return 0;
    }
    return C(n+m, m) - C(n+m, k-1);
  }
  inline T Catalan_s(long long n, long long m, long long k){
    if(k <= 0){
      return C_s(n+m, n);
    }
    if(n < k || m < k){
      return 0;
    }
    return C_s(n+m, m) - C_s(n+m, k-1);
  }
  inline T C_s(long long a, long long b){
    long long i;
    T res;
    if(b < 0 || b > a){
      return 0;
    }
    if(b > a - b){
      b = a - b;
    }
    res = 1;
    for(i=(0);i<(b);i++){
      res *= a - i;
      res /= i + 1;
    }
    return res;
  }
  inline T P_s(long long a, long long b){
    long long i;
    T res;
    if(b < 0 || b > a){
      return 0;
    }
    res = 1;
    for(i=(0);i<(b);i++){
      res *= a - i;
    }
    return res;
  }
  inline T H_s(long long a, long long b){
    if(a==0 && b==0){
      return 1;
    }
    if(a <= 0 || b < 0){
      return 0;
    }
    return C_s(a+b-1, b);
  }
  inline T per_s(long long n, long long k){
    T d;
    int m;
    if(n < 0 || k < 0){
      return 0;
    }
    if(n == k  &&  k == 0){
      return 1;
    }
    if(n == 0 || k == 0){
      return 0;
    }
    if(k==1){
      return 1;
    }
    if(k==2){
      d = n / 2;
      return d;
    }
    if(k==3){
      d = (n-1) / 6;
      m = (n-1) % 6;
      if(m==0){
        return 3 * d * d + d;
      }
      if(m==1){
        return 3 * d * d + 2 * d;
      }
      if(m==2){
        return 3 * d * d + 3 * d + 1;
      }
      if(m==3){
        return 3 * d * d + 4 * d + 1;
      }
      if(m==4){
        return 3 * d * d + 5 * d + 2;
      }
      if(m==5){
        return 3 * d * d + 6 * d + 3;
      }
    }
    assert(0 && "per_s should be k <= 3");
    return -1;
  }
  inline void expand_dfact(int k){
    int i;
    if(k <= mem_dfact){
      return;
    }
    chmax(k, 3);
    chmax(k, 2 * mem_dfact);
    if(mem_dfact==0){
      dfactri = (T*)malloc(k * sizeof(T));
      dfactri[0] = dfactri[1] = 1;
      for(i=(2);i<(k);i++){
        dfactri[i] = i * dfactri[i-2];
      }
    }
    else{
      dfactri = (T*)realloc(dfactri, k * sizeof(T));
      for(i=(mem_dfact);i<(k);i++){
        dfactri[i] = i * dfactri[i-2];
      }
    }
    mem_dfact = k;
  }
  inline void expand_pw2(int k){
    int i;
    if(k <= mem_pw2){
      return;
    }
    chmax(k, 2 * mem_pw2);
    if(mem_pw2==0){
      pw2c = (T*)malloc(k * sizeof(T));
      pw2c[0] = 1;
      for(i=(1);i<(k);i++){
        pw2c[i] = 2 * pw2c[i-1];
      }
    }
    else{
      pw2c = (T*)realloc(pw2c, k * sizeof(T));
      for(i=(mem_pw2);i<(k);i++){
        pw2c[i] = 2 * pw2c[i-1];
      }
    }
    mem_pw2 = k;
  }
  inline void expand_ipw2(int k){
    int i;
    if(k <= mem_ipw2){
      return;
    }
    chmax(k, 2);
    chmax(k, 2 * mem_ipw2);
    if(mem_ipw2==0){
      ipw2c = (T*)malloc(k * sizeof(T));
      ipw2c[0] = 1;
      ipw2c[1] = ipw2c[0] / 2;
      for(i=(1);i<(k);i++){
        ipw2c[i] = ipw2c[1] * ipw2c[i-1];
      }
    }
    else{
      ipw2c = (T*)realloc(ipw2c, k * sizeof(T));
      for(i=(mem_ipw2);i<(k);i++){
        ipw2c[i] = ipw2c[1] * ipw2c[i-1];
      }
    }
    mem_ipw2 = k;
  }
  inline void expand_pw3(int k){
    int i;
    if(k <= mem_pw3){
      return;
    }
    chmax(k, 2 * mem_pw3);
    if(mem_pw3==0){
      pw3c = (T*)malloc(k * sizeof(T));
      pw3c[0] = 1;
      for(i=(1);i<(k);i++){
        pw3c[i] = 3 * pw3c[i-1];
      }
    }
    else{
      pw3c = (T*)realloc(pw3c, k * sizeof(T));
      for(i=(mem_pw3);i<(k);i++){
        pw3c[i] = 3 * pw3c[i-1];
      }
    }
    mem_pw3 = k;
  }
  inline void expand_ipw3(int k){
    int i;
    if(k <= mem_ipw3){
      return;
    }
    chmax(k, 2);
    chmax(k, 2 * mem_ipw3);
    if(mem_ipw3==0){
      ipw3c = (T*)malloc(k * sizeof(T));
      ipw3c[0] = 1;
      ipw3c[1] = ipw3c[0] / 3;
      for(i=(1);i<(k);i++){
        ipw3c[i] = ipw3c[1] * ipw3c[i-1];
      }
    }
    else{
      ipw3c = (T*)realloc(ipw3c, k * sizeof(T));
      for(i=(mem_ipw3);i<(k);i++){
        ipw3c[i] = ipw3c[1] * ipw3c[i-1];
      }
    }
    mem_ipw3 = k;
  }
  inline void expand_pw10(int k){
    int i;
    if(k <= mem_pw10){
      return;
    }
    chmax(k, 2 * mem_pw10);
    if(mem_pw10==0){
      pw10c = (T*)malloc(k * sizeof(T));
      pw10c[0] = 1;
      for(i=(1);i<(k);i++){
        pw10c[i] = 10 * pw10c[i-1];
      }
    }
    else{
      pw10c = (T*)realloc(pw10c, k * sizeof(T));
      for(i=(mem_pw10);i<(k);i++){
        pw10c[i] = 10 * pw10c[i-1];
      }
    }
    mem_pw10 = k;
  }
  inline void expand_ipw10(int k){
    int i;
    if(k <= mem_ipw10){
      return;
    }
    chmax(k, 2);
    chmax(k, 2 * mem_ipw10);
    if(mem_ipw10==0){
      ipw10c = (T*)malloc(k * sizeof(T));
      ipw10c[0] = 1;
      ipw10c[1] = ipw10c[0] / 10;
      for(i=(1);i<(k);i++){
        ipw10c[i] = ipw10c[1] * ipw10c[i-1];
      }
    }
    else{
      ipw10c = (T*)realloc(ipw10c, k * sizeof(T));
      for(i=(mem_ipw10);i<(k);i++){
        ipw10c[i] = ipw10c[1] * ipw10c[i-1];
      }
    }
    mem_ipw10 = k;
  }
  inline void expand_rep1(int k){
    int i;
    if(k <= mem_rep1){
      return;
    }
    chmax(k, 2 * mem_rep1);
    if(mem_rep1==0){
      rep1c = (T*)malloc(k * sizeof(T));
      rep1c[0] = 0;
      for(i=(1);i<(k);i++){
        rep1c[i] = 10 * rep1c[i-1] + 1;
      }
    }
    else{
      rep1c = (T*)realloc(rep1c, k * sizeof(T));
      for(i=(mem_rep1);i<(k);i++){
        rep1c[i] = 10 * rep1c[i-1] + 1;
      }
    }
    mem_rep1 = k;
  }
  inline T dfac(int k){
    if(k >= 0){
      if(mem_dfact < k+1){
        expand_dfact(k+1);
      }
      return dfactri[k];
    }
    if(k==-1){
      return 1;
    }
    k = - k - 2;
    if(k % 4 == 1){
      return 1 / (-dfac(k));
    }
    return 1 / dfac(k);
  }
  inline T pw2(int k){
    if(k >= 0){
      if(mem_pw2 < k+1){
        expand_pw2(k+1);
      }
      return pw2c[k];
    }
    else{
      k = -k;
      if(mem_ipw2 < k+1){
        expand_ipw2(k+1);
      }
      return ipw2c[k];
    }
  }
  inline T pw3(int k){
    if(k >= 0){
      if(mem_pw3 < k+1){
        expand_pw3(k+1);
      }
      return pw3c[k];
    }
    else{
      k = -k;
      if(mem_ipw3 < k+1){
        expand_ipw3(k+1);
      }
      return ipw3c[k];
    }
  }
  inline T pw10(int k){
    if(k >= 0){
      if(mem_pw10 < k+1){
        expand_pw10(k+1);
      }
      return pw10c[k];
    }
    else{
      k = -k;
      if(mem_ipw10 < k+1){
        expand_ipw10(k+1);
      }
      return ipw10c[k];
    }
  }
  inline T repunit(int k){
    if(mem_rep1 < k+1){
      expand_rep1(k+1);
    }
    return rep1c[k];
  }
}
;
template<> inline Modint Comb<Modint>::C_s(long long a, long long b){
  long long i;
  Modint res;
  Modint d;
  if(b < 0 || b > a){
    return 0;
  }
  if(b > a - b){
    b = a - b;
  }
  res = d = 1;
  for(i=(0);i<(b);i++){
    res *= a - i;
    d *= i + 1;
  }
  return res / d;
}
template<class T> int coordcomp_L(int n1, T arr1[], int n2, T arr2[], int n3, T arr3[], int res1[] = NULL, int res2[] = NULL, int res3[] = NULL, void *mem = wmem){
  int i;
  int k = 0;
  int nn = n1 + n2 + n3;
  pair<T,int>*r;
  walloc1d(&r, nn, &mem);
  for(i=(0);i<(n1);i++){
    r[i].first = arr1[i];
    r[i].second = i;
  }
  for(i=(0);i<(n2);i++){
    r[n1+i].first = arr2[i];
    r[n1+i].second = n1+i;
  }
  for(i=(0);i<(n3);i++){
    r[n1+n2+i].first = arr3[i];
    r[n1+n2+i].second = n1+n2+i;
  }
  sort(r, r+nn);
  for(i=(0);i<(nn);i++){
    if(i && r[i].first != r[i-1].first){
      k++;
    }
    if(r[i].second < n1){
      if(res1!=NULL){
        res1[r[i].second] = k;
      }
      else{
        arr1[r[i].second] = k;
      }
    }
    else if(r[i].second < n1+n2){
      if(res2!=NULL){
        res2[r[i].second-n1] = k;
      }
      else{
        arr2[r[i].second-n1] = k;
      }
    }
    else{
      if(res3!=NULL){
        res3[r[i].second-n1-n2] = k;
      }
      else{
        arr3[r[i].second-n1-n2] = k;
      }
    }
  }
  return k+1;
}
template<class T> struct fenwick{
  int size;
  int memory;
  T*data;
  void malloc(int mem);
  void malloc(int mem, int fg);
  void walloc(int mem, void **workMemory = &wmem);
  void walloc(int mem, int fg, void **workMemory = &wmem);
  void free(void);
  void init(int N);
  void add(int k, T val);
  T get(int k);
  T range(int a, int b);
  int kth(T k);
}
;
template<class S, class T1, class T2> struct rangeTree2d{
  int N;
  int N2;
  int*sz;
  S*tot;
  fenwick<S>*w;
  T1**d1;
  T1*ddd1;
  T2*d2;
  inline void build(int nn, T1 dd1[], T2 dd2[], S ww[] = NULL, void **mem = &wmem){
    int i;
    int j;
    int i1;
    int i2;
    int k1;
    int k2;
    int s;
    int s1;
    int s2;
    S*www;
    int*ind;
    N = nn;
    for(N2=1;N2<N;N2*=2){
      ;
    }
    walloc1d(&sz,2*N2,mem);
    walloc1d(&tot,2*N2,mem);
    walloc1d(&w,2*N2,mem);
    walloc1d(&d1,2*N2,mem);
    walloc1d(&d2,nn,mem);
    malloc1d(&www,nn);
    walloc1d(&ddd1,nn);
    walloc1d(&ind,nn);
    for(i=(0);i<(N);i++){
      ddd1[i] = dd1[i];
    }
    for(i=(0);i<(N);i++){
      d2[i] = dd2[i];
    }
    if(ww==NULL){
      for(i=(0);i<(N);i++){
        www[i] = 1;
      }
      sortA_L(N,d2,ddd1);
    }
    else{
      for(i=(0);i<(N);i++){
        ind[i] = i;
      }
      sortA_L(N,d2,ddd1,ind);
      for(i=(0);i<(N);i++){
        www[i] = ww[ind[i]];
      }
    }
    for(i=(0);i<(N);i++){
      sz[N2+i] = 1;
      walloc1d(&d1[N2+i], 1, mem);
      d1[N2+i][0] = ddd1[i];
      w[N2+i].walloc(1, mem);
      w[N2+i].init(1);
      w[N2+i].add(0,www[i]);
      tot[N2+i] = www[i];
    }
    for(i=(N);i<(N2);i++){
      sz[N2+i] = 0;
      tot[N2+i] = 0;
    }
    for(i=(N2)-1;i>=(1);i--){
      i1 = 2*i;
      i2 = 2*i + 1;
      s1 = sz[i1];
      s2 = sz[i2];
      sz[i] = s1 + s2;
      s = k1 = k2 = 0;
      walloc1d(&d1[i], sz[i], mem);
      w[i].walloc(sz[i], mem);
      w[i].init(sz[i]);
      while(k1 < s1 || k2 < s2){
        if(k2==s2){
          d1[i][s] = d1[i1][k1];
          w[i].add(s,w[i1].range(k1,k1));
          s++;
          k1++;
          continue;
        }
        if(k1==s1){
          d1[i][s] = d1[i2][k2];
          w[i].add(s,w[i2].range(k2,k2));
          s++;
          k2++;
          continue;
        }
        if(d1[i1][k1] < d1[i2][k2]){
          d1[i][s] = d1[i1][k1];
          w[i].add(s,w[i1].range(k1,k1));
          s++;
          k1++;
          continue;
        }
        else{
          d1[i][s] = d1[i2][k2];
          w[i].add(s,w[i2].range(k2,k2));
          s++;
          k2++;
          continue;
        }
      }
    }
    free1d(www);
  }
  inline void add(T1 x, T2 y, S v){
    int a;
    int b;
    int z;
    a = lower_bound(d2, d2+N, y) - d2;
    b = upper_bound(d2, d2+N, y) - d2;
    z = lower_bound(ddd1+a, ddd1+b, x) - ddd1 + N2;
    while(z){
      a = lower_bound(d1[z], d1[z]+sz[z], x) - d1[z];
      w[z].add(a, v);
      z /= 2;
    }
  }
  inline S query(T1 x1, T1 x2, T2 y1, T2 y2){
    S res = 0;
    int a;
    int b;
    int z1;
    int z2;
    a = lower_bound(d2, d2+N, y1) - d2 + N2;
    b = lower_bound(d2, d2+N, y2) - d2 + N2;
    while(a < b){
      if(a%2){
        z1 = lower_bound(d1[a], d1[a]+sz[a], x1) - d1[a];
        z2 = lower_bound(d1[a], d1[a]+sz[a], x2) - d1[a];
        if(z1 < z2){
          res += w[a].range(z1,z2-1);
        }
        a++;
      }
      if(b%2){
        b--;
        z1 = lower_bound(d1[b], d1[b]+sz[b], x1) - d1[b];
        z2 = lower_bound(d1[b], d1[b]+sz[b], x2) - d1[b];
        if(z1 < z2){
          res += w[b].range(z1,z2-1);
        }
      }
      a /= 2;
      b /= 2;
    }
    return res;
  }
}
;
long long cReader_ll(long long mn, long long mx, char nx){
  int i;
  int fg = 0;
  int m = 1;
  int f = -1;
  long long res = 0;
  double tmp = 0;
  for(;;){
    i = my_getchar_unlocked();
    if(fg==0 && i=='-'){
      fg++;
      m = -1;
    }
    else if('0' <= i  &&  i <= '9'){
      fg++;
      if(f == -1){
        f = i - '0';
      }
      res = 10 * res + i - '0';
      tmp = 10 * tmp + i - '0';
      assert(tmp < 1e20);
    }
    else{
      break;
    }
  }
  assert(tmp / 2 <= res);
  assert((m==1 && fg >= 1) || (m==-1 && fg >= 2));
  assert(mn <= m * res  &&  m * res <= mx);
  assert(!(res == 0 && m == -1));
  assert(!(res != 0 && f == 0));
  assert(!(res == 0 && fg >= 2));
  assert(i == nx);
  return m * res;
}
int N;
int A[100000+1];
int v[100000+1];
int ind[100000+1];
fenwick<Modint> f1;
fenwick<Modint> f2;
Comb<Modint> comb;
int solve(){
  int Lj4PdHRW;
  int i;
  int j;
  int k;
  Modint res = 0;
  for(Lj4PdHRW=(0);Lj4PdHRW<(2);Lj4PdHRW++){
    for(i=(0);i<(N);i++){
      auto RZTsC2BF = ((A[i]));
      auto FmcKpFmN = (( i));
      v[i]=RZTsC2BF;
      ind[i]=FmcKpFmN;
    }
    sortA_L(N,v,ind);
    f1.walloc(N,1);
    f2.walloc(N,1);
    k = 0;
    for(i=(0);i<(N);i++){
      while(v[k] < v[i]){
        f1.add(ind[k], comb.pw2(ind[k]));
        f2.add(ind[k], comb.pw2(N-1-ind[k]));
        k++;
      }
      res += f1.range(0,ind[i]) * f2.range(ind[i],N-1);
    }
    for(i=(0);i<(N);i++){
      A[i] = -A[i];
    }
  }
  return res;
}
int baka(){
  int mask;
  int s;
  int arr[12];
  Modint res = 0;
  for(mask=(0);mask<(1<<N);mask++){
    int i;
    s = 0;
    for(i=(0);i<(N);i++){
      if(((mask) &(1<<(i)))){
        arr[s++] = A[i];
      }
    }
    for(i=(1);i<(s-1);i++){
      if(arr[i-1] < arr[i] && arr[i] > arr[i+1]){
        res++;
      }
    }
    for(i=(1);i<(s-1);i++){
      if(arr[i-1] > arr[i] && arr[i] < arr[i+1]){
        res++;
      }
    }
  }
  return res;
}
rangeTree2d<Modint,int,int> t1;
rangeTree2d<Modint,int,int> t2;
int xx[100000];
int yy[100000];
Modint ww[100000];
Modint solve2(){
  int i;
  Modint res = 0;
  for(i=(0);i<(N);i++){
    auto XJIcIBrW = ((i));
    auto jPV_0s1p = (( A[i]));
    auto BUotOFBp = (( comb.pw2(i)));
    xx[i]=XJIcIBrW;
    yy[i]=jPV_0s1p;
    ww[i]=BUotOFBp;
  }
  t1.build(N,xx,yy,ww);
  reverse(ww,ww+N);
  t2.build(N,xx,yy,ww);
  for(i=(0);i<(N);i++){
    res += t1.query(0,i,A[i]+1,1073709056) * t2.query(i+1,N,A[i]+1,1073709056);
    res += t1.query(0,i,0,A[i]) * t2.query(i+1,N,0,A[i]);
  }
  return res;
}
template<class S, class T1, class T2> void offlineSumRangeTree2d(int N, T1 D1[], T2 D2[], S W[], int Q, T1 X1[], T1 X2[], T2 Y1[], T2 Y2[], S res[], void *mem = wmem){
  int i;
  int k;
  int sz;
  int*dd1;
  int*xx1;
  int*xx2;
  T2*yy;
  int*ind;
  fenwick<S> f;
  for(i=(0);i<(Q);i++){
    res[i] = 0;
  }
  walloc1d(&dd1, N, &mem);
  walloc1d(&xx1, Q, &mem);
  walloc1d(&xx2, Q, &mem);
  walloc1d(&yy, N+Q+Q, &mem);
  walloc1d(&ind, N+Q+Q, &mem);
  sz =coordcomp_L(N, D1, Q, X1, Q, X2, dd1, xx1, xx2, mem);
  f.walloc(sz, 1, &mem);
  for(i=(0);i<(Q);i++){
    ind[i] = i;
    yy[i] = Y1[i];
  }
  for(i=(0);i<(Q);i++){
    ind[Q+i] = Q+i;
    yy[Q+i] = Y2[i];
  }
  for(i=(0);i<(N);i++){
    ind[Q+Q+i] = Q+Q+i;
    yy[Q+Q+i] = D2[i];
  }
  sortA_L(N+Q+Q, yy, ind, mem);
  for(i=(0);i<(N+Q+Q);i++){
    if(ind[i] < Q){
      k = ind[i];
      if(xx1[k] >= xx2[k] || Y1[k] >= Y2[k]){
        continue;
      }
      res[k] -= f.range(xx1[k], xx2[k]-1);
    }
    else if(ind[i] < Q + Q){
      k = ind[i] - Q;
      if(xx1[k] >= xx2[k] || Y1[k] >= Y2[k]){
        continue;
      }
      res[k] += f.range(xx1[k], xx2[k]-1);
    }
    else{
      k = ind[i] - Q - Q;
      f.add(dd1[k], W[k]);
    }
  }
}
int q;
int xx1[400000];
int yy1[400000];
int xx2[400000];
int yy2[400000];
Modint rrr[400000];
Modint solve3(){
  int i;
  Modint res = 0;
  for(i=(0);i<(N);i++){
    auto o3WxPXbE = ((0));
    auto lQU550vz = (( i));
    auto qE8LMwYZ = (( A[i]+1));
    auto dKuENJNI = (( 1073709056));
    xx1[i+0*N]=o3WxPXbE;
    xx2[i+0*N]=lQU550vz;
    yy1[i+0*N]=qE8LMwYZ;
    yy2[i+0*N]=dKuENJNI;
  }
  for(i=(0);i<(N);i++){
    auto XNa8avth = ((0));
    auto mlGkBPoR = (( i));
    auto YlLMHsfa = (( 0));
    auto sMcf5Tpe = (( A[i]));
    xx1[i+1*N]=XNa8avth;
    xx2[i+1*N]=mlGkBPoR;
    yy1[i+1*N]=YlLMHsfa;
    yy2[i+1*N]=sMcf5Tpe;
  }
  for(i=(0);i<(N);i++){
    auto KaFyNJB9 = ((i+1));
    auto AAsEZMFe = (( N));
    auto xtzQOlbs = (( A[i]+1));
    auto aFgbOQYS = (( 1073709056));
    xx1[i+2*N]=KaFyNJB9;
    xx2[i+2*N]=AAsEZMFe;
    yy1[i+2*N]=xtzQOlbs;
    yy2[i+2*N]=aFgbOQYS;
  }
  for(i=(0);i<(N);i++){
    auto IlgsnSAd = ((i+1));
    auto jG1yfsum = (( N));
    auto NLJcSLph = (( 0));
    auto Wu3kZ3t7 = (( A[i]));
    xx1[i+3*N]=IlgsnSAd;
    xx2[i+3*N]=jG1yfsum;
    yy1[i+3*N]=NLJcSLph;
    yy2[i+3*N]=Wu3kZ3t7;
  }
  for(i=(0);i<(N);i++){
    auto grBCmONb = ((i));
    auto WZu7joIG = (( A[i]));
    auto Wv3_QJ0O = (( comb.pw2(i)));
    xx[i]=grBCmONb;
    yy[i]=WZu7joIG;
    ww[i]=Wv3_QJ0O;
  }
  offlineSumRangeTree2d(N, xx, yy, ww, 2*N, xx1, xx2, yy1, yy2, rrr);
  reverse(ww,ww+N);
  offlineSumRangeTree2d(N, xx, yy, ww, 2*N, xx1+2*N, xx2+2*N, yy1+2*N, yy2+2*N, rrr+2*N);
  for(i=(0);i<(N);i++){
    res += rrr[i+0*N] * rrr[i+2*N];
    res += rrr[i+1*N] * rrr[i+3*N];
  }
  return res;
}
int main(){
  int dtztiDQx;
  wmem = memarr;
  int i;
  int j;
  int k;
  Modint res = 0;
  N = cReader_ll(1, 100000, '\n');
  for(i=(0);i<(N);i++){
    if(i<N-1){
      A[i] = cReader_ll(1, 1000000000,' ');
    }
    else{
      A[i] = cReader_ll(1, 1000000000,'\n');
    }
  }
  wt_L(solve3());
  wt_L('\n');
  return 0;
  for(dtztiDQx=(0);dtztiDQx<(2);dtztiDQx++){
    for(i=(0);i<(N);i++){
      auto memzHCbB = ((A[i]));
      auto mRLrOZDI = (( i));
      v[i]=memzHCbB;
      ind[i]=mRLrOZDI;
    }
    sortA_L(N,v,ind);
    f1.walloc(N,1);
    f2.walloc(N,1);
    k = 0;
    for(i=(0);i<(N);i++){
      while(v[k] < v[i]){
        f1.add(ind[k], comb.pw2(ind[k]));
        f2.add(ind[k], comb.pw2(N-1-ind[k]));
        k++;
      }
      res += f1.range(0,ind[i]) * f2.range(ind[i],N-1);
    }
    for(i=(0);i<(N);i++){
      A[i] = -A[i];
    }
  }
  wt_L(res);
  wt_L('\n');
  return 0;
}
template<class T> void fenwick<T>::malloc(int mem){
  memory = mem;
  data = (T*)std::malloc(sizeof(T)*mem);
}
template<class T> void fenwick<T>::malloc(int mem, int fg){
  memory = mem;
  data = (T*)std::malloc(sizeof(T)*mem);
  if(fg){
    init(mem);
  }
}
template<class T> void fenwick<T>::walloc(int mem, void **workMemory /* = &wmem*/){
  memory = mem;
  walloc1d(&data, mem, workMemory);
}
template<class T> void fenwick<T>::walloc(int mem, int fg, void **workMemory /* = &wmem*/){
  memory = mem;
  walloc1d(&data, mem, workMemory);
  if(fg){
    init(mem);
  }
}
template<class T> void fenwick<T>::free(void){
  memory = 0;
  free(data);
}
template<class T> void fenwick<T>::init(int N){
  size = N;
  memset(data,0,sizeof(T)*N);
}
template<class T> void fenwick<T>::add(int k, T val){
  while(k < size){
    data[k] += val;
    k |= k+1;
  }
}
template<class T> T fenwick<T>::get(int k){
  T res = 0;
  while(k>=0){
    res += data[k];
    k = (k&(k+1))-1;
  }
  return res;
}
template<class T> T fenwick<T>::range(int a, int b){
  if(a < 0){
    a = 0;
  }
  if(b >= size){
    b = size - 1;
  }
  if(b < a){
    return 0;
  }
  return get(b) - get(a-1);
}
template<class T> int fenwick<T>::kth(T k){
  int i=0;
  int j=size;
  int c;
  T v;
  while(i<j){
    c = (i+j)/2;
    v = get(c);
    if(v <= k){
      i=c+1;
    }
    else{
      j=c;
    }
  }
  return i==size?-1:i;
}
// cLay version 20210819-1 [beta]

// --- original code ---
// //working_memory=300m
// int N, A[1d5+1], v[], ind[];
// fenwick<Modint> f1, f2;
// Comb<Modint> comb;
// 
// int solve(){
//   int i, j, k;
//   Modint res = 0;
//   rep(2){
//     rep(i,N) (v[i], ind[i]) = (A[i], i);
//     sortA(N,v,ind);
// 
//     f1.walloc(N,1);
//     f2.walloc(N,1);
// 
//     k = 0;
//     rep(i,N){
//       while(v[k] < v[i]){
//         f1.add(ind[k], comb.pw2(ind[k]));
//         f2.add(ind[k], comb.pw2(N-1-ind[k]));
//         k++;
//       }
//       res += f1.range(0,ind[i]) * f2.range(ind[i],N-1);
//     }
// 
//     rep(i,N) A[i] = -A[i];
//   }
//   return res;
// }
// 
// int baka(){
//   int s, arr[12];
//   Modint res = 0;
// 
//   rep(mask,1<<N){
//     s = 0;
//     rep(i,N) if(BIT_ith(mask,i)) arr[s++] = A[i];
//     rep(i,1,s-1) if(arr[i-1] < arr[i] && arr[i] > arr[i+1]) res++;
//     rep(i,1,s-1) if(arr[i-1] > arr[i] && arr[i] < arr[i+1]) res++;
//   }
// 
//   return res;
// }
// 
// rangeTree2d<Modint,int,int> t1, t2;
// int xx[1d5], yy[1d5]; Modint ww[1d5];
// Modint solve2(){
//   Modint res = 0;
//   rep(i,N) (xx[i], yy[i], ww[i]) = (i, A[i], comb.pw2(i));
//   t1.build(N,xx,yy,ww);
//   reverse(ww,ww+N);
//   t2.build(N,xx,yy,ww);
//   rep(i,N){
//     res += t1.query(0,i,A[i]+1,int_inf) * t2.query(i+1,N,A[i]+1,int_inf);
//     res += t1.query(0,i,0,A[i]) * t2.query(i+1,N,0,A[i]);
//   }
//   return res;
// }
// 
// 
// 
// template<class S, class T1, class T2>
// void offlineSumRangeTree2d(int N, T1 D1[], T2 D2[], S W[], int Q, T1 X1[], T1 X2[], T2 Y1[], T2 Y2[], S res[], void *mem = wmem){
//   int i, k;
//   int sz, *dd1, *xx1, *xx2;
//   T2 *yy; int *ind;
//   fenwick<S> f;
// 
//   rep(i,Q) res[i] = 0;
// 
//   walloc1d(&dd1, N, &mem);
//   walloc1d(&xx1, Q, &mem);
//   walloc1d(&xx2, Q, &mem);
//   walloc1d(&yy, N+Q+Q, &mem);
//   walloc1d(&ind, N+Q+Q, &mem);
//   sz = coordcomp(N, D1, Q, X1, Q, X2, dd1, xx1, xx2, mem);
// 
//   f.walloc(sz, 1, &mem);
//   rep(i,Q) ind[i] = i, yy[i] = Y1[i];
//   rep(i,Q) ind[Q+i] = Q+i, yy[Q+i] = Y2[i];
//   rep(i,N) ind[Q+Q+i] = Q+Q+i, yy[Q+Q+i] = D2[i];
//   sortA(N+Q+Q, yy, ind, mem);
// 
//   rep(i,N+Q+Q){
//     if(ind[i] < Q){
//       k = ind[i];
//       if(xx1[k] >= xx2[k] || Y1[k] >= Y2[k]) continue;
//       res[k] -= f.range(xx1[k], xx2[k]-1);
//     } else if(ind[i] < Q + Q){
//       k = ind[i] - Q;
//       if(xx1[k] >= xx2[k] || Y1[k] >= Y2[k]) continue;
//       res[k] += f.range(xx1[k], xx2[k]-1);
//     } else {
//       k = ind[i] - Q - Q;
//       f.add(dd1[k], W[k]);
//     }
//   }
// }
// 
// 
// int q, xx1[4d5], yy1[], xx2[], yy2[]; Modint rrr[];
// Modint solve3(){
//   Modint res = 0;
//   rep(i,N) (xx1[i+0*N], xx2[i+0*N], yy1[i+0*N], yy2[i+0*N]) = (0, i, A[i]+1, int_inf);
//   rep(i,N) (xx1[i+1*N], xx2[i+1*N], yy1[i+1*N], yy2[i+1*N]) = (0, i, 0, A[i]);
//   rep(i,N) (xx1[i+2*N], xx2[i+2*N], yy1[i+2*N], yy2[i+2*N]) = (i+1, N, A[i]+1, int_inf);
//   rep(i,N) (xx1[i+3*N], xx2[i+3*N], yy1[i+3*N], yy2[i+3*N]) = (i+1, N, 0, A[i]);
// 
//   rep(i,N) (xx[i], yy[i], ww[i]) = (i, A[i], comb.pw2(i));
//   offlineSumRangeTree2d(N, xx, yy, ww, 2*N, xx1, xx2, yy1, yy2, rrr);
//   reverse(ww,ww+N);
//   offlineSumRangeTree2d(N, xx, yy, ww, 2*N, xx1+2*N, xx2+2*N, yy1+2*N, yy2+2*N, rrr+2*N);
//   rep(i,N){
//     res += rrr[i+0*N] * rrr[i+2*N];
//     res += rrr[i+1*N] * rrr[i+3*N];
//   }
//   return res;
// }
// 
// {
//   int i, j, k;
//   Modint res = 0;
//   N = cReader_ll(1, 1d5, '\n');
//   rep(i,N) A[i] = cReader_ll(1, 1d9, if[i<N-1, ' ', '\n']);
// 
//   wt(solve3());
//   return 0;
// 
//   // Rand rnd;
//   // for(;;){
//   //   int res1, res2;
//   //   void *mem = wmem;
//   //   N = rnd.get(1,11);
//   //   rep(i,N) A[i] = rnd.get(1,10);
//   //   res1 = solve();
//   //   res2 = baka();
//   //   printf("%d %d %d\n",N,res1,res2);
//   //   assert(res1==res2);
//   //   wmem = mem;
//   // }
// 
//   rep(2){
//     rep(i,N) (v[i], ind[i]) = (A[i], i);
//     sortA(N,v,ind);
// 
//     f1.walloc(N,1);
//     f2.walloc(N,1);
// 
//     k = 0;
//     rep(i,N){
//       while(v[k] < v[i]){
//         f1.add(ind[k], comb.pw2(ind[k]));
//         f2.add(ind[k], comb.pw2(N-1-ind[k]));
//         k++;
//       }
//       res += f1.range(0,ind[i]) * f2.range(ind[i],N-1);
//     }
// 
//     rep(i,N) A[i] = -A[i];
//   }
// 
//   wt(res);
// }