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#pragma GCC optimize ("Ofast")
#include<bits/stdc++.h>
using namespace std;
#define MD (1000000007U)
template<class T> struct cLtraits_identity{
  using type = T;
}
;
template<class T> using cLtraits_try_make_signed =
  typename conditional<
    is_integral<T>::value,
    make_signed<T>,
    cLtraits_identity<T>
    >::type;
template <class S, class T> struct cLtraits_common_type{
  using tS = typename cLtraits_try_make_signed<S>::type;
  using tT = typename cLtraits_try_make_signed<T>::type;
  using type = typename common_type<tS,tT>::type;
}
;
void*wmem;
char memarr[96000000];
template<class S, class T> inline auto min_L(S a, T b)
-> typename cLtraits_common_type<S,T>::type{
  return (typename cLtraits_common_type<S,T>::type) a <= (typename cLtraits_common_type<S,T>::type) b ? a : b;
}
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;
}
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++];
}
inline void rd(int &x){
  int k;
  int m=0;
  x=0;
  for(;;){
    k = my_getchar_unlocked();
    if(k=='-'){
      m=1;
      break;
    }
    if('0'<=k&&k<='9'){
      x=k-'0';
      break;
    }
  }
  for(;;){
    k = my_getchar_unlocked();
    if(k<'0'||k>'9'){
      break;
    }
    x=x*10+k-'0';
  }
  if(m){
    x=-x;
  }
}
inline void rd(long long &x){
  int k;
  int m=0;
  x=0;
  for(;;){
    k = my_getchar_unlocked();
    if(k=='-'){
      m=1;
      break;
    }
    if('0'<=k&&k<='9'){
      x=k-'0';
      break;
    }
  }
  for(;;){
    k = my_getchar_unlocked();
    if(k<'0'||k>'9'){
      break;
    }
    x=x*10+k-'0';
  }
  if(m){
    x=-x;
  }
}
inline int rd_int(void){
  int x;
  rd(x);
  return x;
}
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 1;
  }
  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 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 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> struct segtree{
  int N;
  int logN;
  T*sum;
  T*mn;
  int*mnind;
  T*fixval;
  char*fixed;
  T*addval;
  void malloc(int maxN, int once = 0){
    int i;
    for(i=1;i<maxN;i*=2){
      ;
    }
    sum = new T[2*i];
    mn = new T[2*i];
    mnind = new int[2*i];
    fixval = new T[i];
    addval = new T[i];
    fixed = new char[i];
    if(once){
      setN(maxN);
    }
  }
  void walloc(int maxN, int once = 0, void **mem = &wmem){
    int i;
    for(i=1;i<maxN;i*=2){
      ;
    }
    walloc1d(&sum, 2*i, mem);
    walloc1d(&mn, 2*i, mem);
    walloc1d(&mnind, 2*i, mem);
    walloc1d(&fixval, i, mem);
    walloc1d(&addval, i, mem);
    walloc1d(&fixed, i, mem);
    if(once){
      setN(maxN);
    }
  }
  void free(void){
    delete [] sum;
    delete [] mn;
    delete [] mnind;
    delete [] fixval;
    delete [] addval;
    delete [] fixed;
  }
  T& operator[](int i){
    return sum[N+i];
  }
  void setN(int n, int zerofill = 1, int dobuild = 1){
    int i;
    for(i=1,logN=0;i<n;i*=2,logN++){
      ;
    }
    N = i;
    if(zerofill){
      for(i=(0);i<(N);i++){
        sum[N+i] = 0;
      }
    }
    if(dobuild){
      build();
    }
  }
  void build(void){
    int i;
    for(i=(0);i<(N);i++){
      mn[N+i] = sum[N+i];
      mnind[N+i] = i;
    }
    for(i=N-1;i;i--){
      sum[i] = sum[2*i] + sum[2*i+1];
      if(mn[2*i] <= mn[2*i+1]){
        mn[i] = mn[2*i];
        mnind[i] = mnind[2*i];
      }
      else{
        mn[i] = mn[2*i+1];
        mnind[i] = mnind[2*i+1];
      }
    }
    int g65CQCoC = N;
    for(i=(1);i<(g65CQCoC);i++){
      fixed[i] = 0;
    }
    int emd5LSgV = N;
    for(i=(1);i<(emd5LSgV);i++){
      addval[i] = 0;
    }
  }
  inline void push_one(int a, int sz, int st){
    if(fixed[a]){
      if(sz > 1){
        fixed[a*2] = fixed[a*2+1] = 1;
        fixval[a*2] = fixval[a*2+1] = fixval[a];
        sum[a*2] = sum[a*2+1] = sz * fixval[a];
        mn[a*2] = mn[a*2+1] = fixval[a];
        mnind[a*2] = st;
        mnind[a*2+1] = st + sz;
      }
      else{
        sum[a*2] = sum[a*2+1] = sz * fixval[a];
        mn[a*2] = mn[a*2+1] = fixval[a];
        mnind[a*2] = st;
        mnind[a*2+1] = st + sz;
      }
      fixed[a] = 0;
      addval[a] = 0;
      return;
    }
    if(addval[a] != 0){
      if(sz > 1){
        if(fixed[a*2]){
          fixval[a*2] += addval[a];
        }
        else{
          addval[a*2] += addval[a];
        }
        if(fixed[a*2+1]){
          fixval[a*2+1] += addval[a];
        }
        else{
          addval[a*2+1] += addval[a];
        }
        sum[a*2] += sz * addval[a];
        sum[a*2+1] += sz * addval[a];
        mn[a*2] += addval[a];
        mn[a*2+1] += addval[a];
      }
      else{
        sum[a*2] += sz * addval[a];
        sum[a*2+1] += sz * addval[a];
        mn[a*2] += addval[a];
        mn[a*2+1] += addval[a];
      }
      addval[a] = 0;
      return;
    }
  }
  inline void push(int a){
    int i;
    int aa = a - N;
    int nd;
    int sz;
    int st;
    for(i=logN;i;i--){
      nd = a>>i;
      sz = 1<<(i-1);
      st = 2 * sz * (aa>>i);
      push_one(nd, sz, st);
    }
  }
  inline void build(int a){
    int sz = 1;
    int st = a - N;
    while(a > 1){
      if(a%2){
        st += sz;
      }
      a /= 2;
      sz *= 2;
      if(fixed[a]){
        sum[a] = sz * fixval[a];
        mn[a] = fixval[a];
      }
      else{
        sum[a] = sum[a*2] + sum[a*2+1];
        if(mn[a*2] <= mn[a*2+1]){
          mn[a] = mn[a*2];
          mnind[a] = mnind[a*2];
        }
        else{
          mn[a] = mn[a*2+1];
          mnind[a] = mnind[a*2+1];
        }
        if(addval[a] != 0){
          mn[a] += addval[a];
          sum[a] += sz * addval[a];
        }
      }
    }
  }
  inline void change(int a, int b, T val){
    int sz = 1;
    int aa;
    int bb;
    int st_a = a;
    int st_b = b;
    if(a >= b){
      return;
    }
    aa = (a += N);
    bb = (b += N);
    push(a);
    push(b-1);
    if(a%2){
      sum[a] = mn[a] = val;
      a++;
      st_a += sz;
    }
    if(b%2){
      b--;
      st_b -= sz;
      sum[b] = mn[b] = val;
    }
    a /= 2;
    b /= 2;
    while(a < b){
      sz *= 2;
      if(a%2){
        fixed[a]=1;
        fixval[a]=val;
        sum[a] = sz * val;
        mn[a] = val;
        mnind[a] = st_a;
        a++;
        st_a += sz;
      }
      if(b%2){
        b--;
        st_b -= sz;
        fixed[b]=1;
        fixval[b]=val;
        sum[b] = sz * val;
        mn[b] = val;
        mnind[b] = st_b;
      }
      a /= 2;
      b /= 2;
    }
    build(aa);
    build(bb-1);
  }
  inline void add(int a, int b, T val){
    int sz = 1;
    int aa;
    int bb;
    if(a >= b){
      return;
    }
    aa = (a += N);
    bb = (b += N);
    push(a);
    push(b-1);
    if(a%2){
      sum[a] += val;
      mn[a] += val;
      a++;
    }
    if(b%2){
      b--;
      sum[b] += val;
      mn[b] += val;
    }
    a /= 2;
    b /= 2;
    while(a < b){
      sz *= 2;
      if(a%2){
        if(fixed[a]){
          fixval[a] += val;
        }
        else{
          addval[a] += val;
        }
        sum[a] += sz * val;
        mn[a] += val;
        a++;
      }
      if(b%2){
        b--;
        if(fixed[b]){
          fixval[b] += val;
        }
        else{
          addval[b] += val;
        }
        sum[b] += sz * val;
        mn[b] += val;
      }
      a /= 2;
      b /= 2;
    }
    build(aa);
    build(bb-1);
  }
  inline pair<T,int> getMin(int a, int b){
    pair<T,int> res;
    pair<T,int> tmp;
    int fga = 0;
    int fgb = 0;
    a += N;
    b += N;
    push(a);
    push(b-1);
    while(a < b){
      if(a%2){
        if(fga){
          res =min_L(res, make_pair(mn[a], mnind[a]));
        }
        else{
          res = make_pair(mn[a], mnind[a]);
          fga = 1;
        }
        a++;
      }
      if(b%2){
        b--;
        if(fgb){
          tmp =min_L(make_pair(mn[b], mnind[b]), tmp);
        }
        else{
          tmp = make_pair(mn[b], mnind[b]);
          fgb = 1;
        }
      }
      a /= 2;
      b /= 2;
    }
    if(fga==1 && fgb==0){
      return res;
    }
    if(fga==0 && fgb==1){
      return tmp;
    }
    if(fga==1 && fgb==1){
      res =min_L(res, tmp);
      return res;
    }
    return res;
  }
  inline T getMinVal(int a, int b){
    T res;
    T tmp;
    int fga = 0;
    int fgb = 0;
    a += N;
    b += N;
    push(a);
    push(b-1);
    while(a < b){
      if(a%2){
        if(fga){
          res =min_L(res, mn[a]);
        }
        else{
          res = mn[a];
          fga = 1;
        }
        a++;
      }
      if(b%2){
        b--;
        if(fgb){
          tmp =min_L(mn[b], tmp);
        }
        else{
          tmp = mn[b];
          fgb = 1;
        }
      }
      a /= 2;
      b /= 2;
    }
    if(fga==1 && fgb==0){
      return res;
    }
    if(fga==0 && fgb==1){
      return tmp;
    }
    if(fga==1 && fgb==1){
      res =min_L(res, tmp);
      return res;
    }
    return res;
  }
  inline int getMinInd(int a, int b){
    return getMin(a,b).second;
  }
  inline T getSum(int a, int b){
    T res;
    T tmp;
    int fga = 0;
    int fgb = 0;
    a += N;
    b += N;
    push(a);
    push(b-1);
    while(a < b){
      if(a%2){
        if(fga){
          res = res + sum[a];
        }
        else{
          res = sum[a];
          fga = 1;
        }
        a++;
      }
      if(b%2){
        b--;
        if(fgb){
          tmp = sum[b] + tmp;
        }
        else{
          tmp = sum[b];
          fgb = 1;
        }
      }
      a /= 2;
      b /= 2;
    }
    if(fga==1 && fgb==0){
      return res;
    }
    if(fga==0 && fgb==1){
      return tmp;
    }
    if(fga==1 && fgb==1){
      res = res + tmp;
      return res;
    }
    return res;
  }
}
;
int TEST;
int N;
long long A[100000+1];
Comb<Modint> c;
segtree<long long> t;
Modint solve(int x, int y){
  int z;
  if(x >= y){
    return 1;
  }
  z = t.getMinInd(x, y);
  return solve(x,z) * solve(z+1,y) * c.C(y-x-1, z-x);
}
int main(){
  wmem = memarr;
  t.walloc(100000+1);
  int Lj4PdHRW = rd_int();
  for(TEST=(0);TEST<(Lj4PdHRW);TEST++){
    int i;
    wt_L("Case #");
    wt_L(TEST+1);
    wt_L(": ");
    rd(N);
    {
      int e98WHCEY;
      for(e98WHCEY=(0);e98WHCEY<(N);e98WHCEY++){
        rd(A[e98WHCEY]);
      }
    }
    for(i=(1);i<(N);i++){
      if(A[i] > A[i-1]+1){
        wt_L(0);
        wt_L('\n');
        goto KL2GvlyY;
      }
    }
    t.setN(N);
    for(i=(0);i<(N);i++){
      t[i] = A[i] * 1000000000 - i;
    }
    t.build();
    wt_L(solve(0, N));
    wt_L('\n');
    KL2GvlyY:;
  }
  return 0;
}
// cLay version 20210405-1
 
// --- original code ---
// int TEST;
// int N; ll A[1d5+1];
// Comb<Modint> c;
// segtree<ll> t;
// 
// Modint solve(int x, int y){
//   int z;
//   if(x >= y) return 1;
//   z = t.getMinInd(x, y);
//   return solve(x,z) * solve(z+1,y) * c.C(y-x-1, z-x);
// }
// 
// {
//   t.walloc(1d5+1);
//   REP(TEST,rd_int()){
//     wtF("Case #{TEST+1}: ");
//     rd(N,A(N));
//     rep(i,1,N) if(A[i] > A[i-1]+1) wt(0), break_continue;
//     t.setN(N);
//     rep(i,N) t[i] = A[i] * 1d9 - i;
//     t.build();
//     wt(solve(0, N));
//   }
// }