#pragma GCC optimize ("Ofast")
#include<bits/stdc++.h>
using namespace std;
#define MD (998244353U)
#define MD_PRIMITIVE_ROOT (3U)
#define PI 3.14159265358979323846
void*wmem;
char memarr[209715200];
template<class S, class T> inline S max_L(S a,T b){
  return a>=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 T1> void sortA_L(int N, T1 a[], void *mem = wmem){
  sort(a, a+N);
}
template<class T1> void rsortA_L(int N, T1 a[], void *mem = wmem){
  sortA_L(N, a, mem);
  reverse(a, a+N);
}
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+=(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;
  }
}
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(Modint x){
  int i;
  i = (int)x;
  wt_L(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;
  Comb(){
    mem_fact = 0;
  }
  inline void expand_fact(int k){
    if(k <= mem_fact){
      return;
    }
    chmax(k, 2* mem_fact);
    if(mem_fact == 0){
      int i;
      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{
      int i;
      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;
  }
}
;
template <class T> struct LHeap{
  int*hp;
  int*place;
  int size;
  T*val;
  void malloc(int N){
    hp = (int*)std::malloc(N*sizeof(int));
    place=(int*)std::malloc(N*sizeof(int));
    val=(T*)std::malloc(N*sizeof(T));
  }
  void walloc(int N, void **mem=&wmem){
    walloc1d(&hp, N, mem);
    walloc1d(&place, N, mem);
    walloc1d(&val, N, mem);
  }
  void free(){
    std::free(hp);
    std::free(place);
    std::free(val);
  }
  void init(int N){
    int i;
    size=0;
    for(i=(0);i<(N);i++){
      place[i]=-1;
    }
  }
  void up(int n){
    int m;
    while(n){
      m=(n-1)/2;
      if(val[hp[m]]<=val[hp[n]]){
        break;
      }
      swap(hp[m],hp[n]);
      swap(place[hp[m]],place[hp[n]]);
      n=m;
    }
  }
  void down(int n){
    int m;
    for(;;){
      m=2*n+1;
      if(m>=size){
        break;
      }
      if(m+1<size&&val[hp[m]]>val[hp[m+1]]){
        m++;
      }
      if(val[hp[m]]>=val[hp[n]]){
        break;
      }
      swap(hp[m],hp[n]);
      swap(place[hp[m]],place[hp[n]]);
      n=m;
    }
  }
  void change(int n, T v){
    T f = val[n];
    val[n] = v;
    if(place[n]==-1){
      place[n] = size;
      hp[size++] = n;
      up(place[n]);
    }
    else{
      if(f < v){
        down(place[n]);
      }
      else if(f > v){
        up(place[n]);
      }
    }
  }
  int pop(void){
    int res = hp[0];
    place[res] = -1;
    size--;
    if(size){
      hp[0]=hp[size];
      place[hp[0]]=0;
      down(0);
    }
    return res;
  }
}
;
struct fft_pnt{
  double x;
  double y;
  fft_pnt(void){
  }
  fft_pnt(double a, double b){
    x = a;
    y = b;
  }
  void set(double a, double b){
    x = a;
    y = b;
  }
  fft_pnt& operator+=(fft_pnt a){
    x+=a.x;
    y+=a.y;
    return *this;
  }
  fft_pnt& operator-=(fft_pnt a){
    x-=a.x;
    y-=a.y;
    return *this;
  }
  fft_pnt& operator*=(fft_pnt a){
    fft_pnt p = *this;
    x = p.x*a.x-p.y*a.y;
    y = p.x*a.y+p.y*a.x;
    return *this;
  }
  fft_pnt operator+(fft_pnt a){
    return fft_pnt(*this) += a;
  }
  fft_pnt operator-(fft_pnt a){
    return fft_pnt(*this) -= a;
  }
  fft_pnt operator*(fft_pnt a){
    return fft_pnt(*this) *= a;
  }
}
;
void fft_L(int n, fft_pnt x[], void *mem){
  int i;
  int j;
  int n1;
  int n2;
  int n3;
  int step = 1;
  double theta = 2*PI / n;
  double tmp;
  fft_pnt w1;
  fft_pnt w2;
  fft_pnt w3;
  fft_pnt a;
  fft_pnt b;
  fft_pnt c;
  fft_pnt d;
  fft_pnt aa;
  fft_pnt bb;
  fft_pnt cc;
  fft_pnt dd;
  fft_pnt*y = (fft_pnt*)mem;
  while(n > 2){
    n1 = n / 4;
    n2 = n1 + n1;
    n3 = n1 + n2;
    for(i=(0);i<(n1);i++){
      w1 = fft_pnt(cos(i*theta),-sin(i*theta));
      w2 = w1*w1;
      w3 = w1*w2;
      for(j=(0);j<(step);j++){
        a = x[j+step*i];
        b = x[j+step*(i+n1)];
        c = x[j+step*(i+n2)];
        d = x[j+step*(i+n3)];
        aa = a + c;
        bb = a - c;
        cc = b + d;
        dd = b - d;
        tmp = dd.y;
        dd.y = dd.x;
        dd.x = -tmp;
        y[j+step*(4*i  )] = aa + cc;
        y[j+step*(4*i+1)] = w1*(bb - dd);
        y[j+step*(4*i+2)] = w2*(aa - cc);
        y[j+step*(4*i+3)] = w3*(bb + dd);
      }
    }
    n /= 4;
    step *= 4;
    theta *= 4;
    swap(x,y);
  }
  if(n==2){
    for(i=(0);i<(step);i++){
      y[i] = x[i] + x[i+step];
      y[i+step] = x[i] - x[i+step];
    }
    n /= 2;
    step *= 2;
    theta *= 2;
    swap(x,y);
  }
  for(i=(0);i<(step);i++){
    y[i] = x[i];
  }
}
void fftinv_L(int n, fft_pnt x[], void *mem){
  int i;
  int j;
  int n1;
  int n2;
  int n3;
  int step = 1;
  double theta = 2*PI / n;
  double tmp;
  fft_pnt w1;
  fft_pnt w2;
  fft_pnt w3;
  fft_pnt a;
  fft_pnt b;
  fft_pnt c;
  fft_pnt d;
  fft_pnt aa;
  fft_pnt bb;
  fft_pnt cc;
  fft_pnt dd;
  fft_pnt*y = (fft_pnt*)mem;
  while(n > 2){
    n1 = n / 4;
    n2 = n1 + n1;
    n3 = n1 + n2;
    for(i=(0);i<(n1);i++){
      w1 = fft_pnt(cos(i*theta),sin(i*theta));
      w2 = w1*w1;
      w3 = w1*w2;
      for(j=(0);j<(step);j++){
        a = x[j+step*i];
        b = x[j+step*(i+n1)];
        c = x[j+step*(i+n2)];
        d = x[j+step*(i+n3)];
        aa = a + c;
        bb = a - c;
        cc = b + d;
        dd = b - d;
        tmp = dd.y;
        dd.y = dd.x;
        dd.x = -tmp;
        y[j+step*(4*i  )] = aa + cc;
        y[j+step*(4*i+1)] = w1*(bb + dd);
        y[j+step*(4*i+2)] = w2*(aa - cc);
        y[j+step*(4*i+3)] = w3*(bb - dd);
      }
    }
    n /= 4;
    step *= 4;
    theta *= 4;
    swap(x,y);
  }
  if(n==2){
    for(i=(0);i<(step);i++){
      y[i] = x[i] + x[i+step];
      y[i+step] = x[i] - x[i+step];
    }
    n /= 2;
    step *= 2;
    theta *= 2;
    swap(x,y);
  }
  for(i=(0);i<(step);i++){
    y[i] = x[i];
  }
}
void convolution_L(double A[], int As, double B[], int Bs, double res[], int Rs, void *mem = wmem){
  int i;
  int n;
  int n2;
  double mul;
  fft_pnt*a;
  fft_pnt*b;
  n =max_L(As+Bs, Rs);
  for(n2=1;n2<n;n2*=2){
    ;
  }
  walloc1d(&a, n2, &mem);
  walloc1d(&b, n2, &mem);
  for(i=(0);i<(As);i++){
    a[i].set(A[i], 0);
  }
  int Nwx7Bhnq = n2;
  for(i=(As);i<(Nwx7Bhnq);i++){
    a[i].set(0,0);
  }
  for(i=(0);i<(Bs);i++){
    b[i].set(B[i], 0);
  }
  int MNlRFaTx = n2;
  for(i=(Bs);i<(MNlRFaTx);i++){
    b[i].set(0,0);
  }
  fft_L(n2, a, mem);
  fft_L(n2, b, mem);
  for(i=(0);i<(n2);i++){
    a[i] *= b[i];
  }
  fftinv_L(n2, a, mem);
  mul = 1.0 / n2;
  for(i=(0);i<(Rs);i++){
    res[i] = a[i].x * mul;
  }
}
void convolution_L(double A[], int As, double res[], int Rs, void *mem = wmem){
  int i;
  int n;
  int n2;
  double mul;
  fft_pnt*a;
  n =max_L(As+As, Rs);
  for(n2=1;n2<n;n2*=2){
    ;
  }
  walloc1d(&a, n2, &mem);
  for(i=(0);i<(As);i++){
    a[i].set(A[i], 0);
  }
  int vzsjNTAU = n2;
  for(i=(As);i<(vzsjNTAU);i++){
    a[i].set(0,0);
  }
  fft_L(n2, a, mem);
  for(i=(0);i<(n2);i++){
    a[i] *= a[i];
  }
  fftinv_L(n2, a, mem);
  mul = 1.0 / n2;
  for(i=(0);i<(Rs);i++){
    res[i] = a[i].x * mul;
  }
}
void fft_L(int n, Modint x[], Modint root, void *mem){
  int i;
  int j;
  int n1;
  int n2;
  int n3;
  int step = 1;
  Modint w1;
  Modint w2;
  Modint w3;
  Modint a;
  Modint b;
  Modint c;
  Modint d;
  Modint aa;
  Modint bb;
  Modint cc;
  Modint dd;
  Modint tmp;
  Modint*y;
  walloc1d(&y, n, &mem);
  tmp = root.pw((MD-1)/4*3);
  root = root.pw((MD-1)/n);
  while(n > 2){
    n1 = n / 4;
    n2 = n1 + n1;
    n3 = n1 + n2;
    w1.val = 1;
    for(i=(0);i<(n1);i++){
      w2 = w1*w1;
      w3 = w1*w2;
      for(j=(0);j<(step);j++){
        a = x[j+step*i];
        b = x[j+step*(i+n1)];
        c = x[j+step*(i+n2)];
        d = x[j+step*(i+n3)];
        aa = a + c;
        bb = a - c;
        cc = b + d;
        dd = (b - d) * tmp;
        y[j+step*(4*i  )] = aa + cc;
        y[j+step*(4*i+1)] = w1*(bb - dd);
        y[j+step*(4*i+2)] = w2*(aa - cc);
        y[j+step*(4*i+3)] = w3*(bb + dd);
      }
      w1 *= root;
    }
    n /= 4;
    step *= 4;
    root *= root;
    root *= root;
    swap(x,y);
  }
  if(n==2){
    for(i=(0);i<(step);i++){
      y[i] = x[i] + x[i+step];
      y[i+step] = x[i] - x[i+step];
    }
    n /= 2;
    step *= 2;
    root *= root;
    swap(x,y);
  }
  for(i=(0);i<(step);i++){
    y[i] = x[i];
  }
}
void fftinv_L(int n, Modint x[], Modint root, void *mem){
  int i;
  int j;
  int n1;
  int n2;
  int n3;
  int step = 1;
  Modint w1;
  Modint w2;
  Modint w3;
  Modint a;
  Modint b;
  Modint c;
  Modint d;
  Modint aa;
  Modint bb;
  Modint cc;
  Modint dd;
  Modint tmp;
  Modint*y;
  walloc1d(&y, n, &mem);
  root = root.inverse();
  tmp = root.pw((MD-1)/4);
  root = root.pw((MD-1)/n);
  while(n > 2){
    n1 = n / 4;
    n2 = n1 + n1;
    n3 = n1 + n2;
    w1.val = 1;
    for(i=(0);i<(n1);i++){
      w2 = w1*w1;
      w3 = w1*w2;
      for(j=(0);j<(step);j++){
        a = x[j+step*i];
        b = x[j+step*(i+n1)];
        c = x[j+step*(i+n2)];
        d = x[j+step*(i+n3)];
        aa = a + c;
        bb = a - c;
        cc = b + d;
        dd = (b - d) * tmp;
        y[j+step*(4*i  )] = aa + cc;
        y[j+step*(4*i+1)] = w1*(bb + dd);
        y[j+step*(4*i+2)] = w2*(aa - cc);
        y[j+step*(4*i+3)] = w3*(bb - dd);
      }
      w1 *= root;
    }
    n /= 4;
    step *= 4;
    root *= root;
    root *= root;
    swap(x,y);
  }
  if(n==2){
    for(i=(0);i<(step);i++){
      y[i] = x[i] + x[i+step];
      y[i+step] = x[i] - x[i+step];
    }
    n /= 2;
    step *= 2;
    root *= root;
    swap(x,y);
  }
  for(i=(0);i<(step);i++){
    y[i] = x[i];
  }
}
void convolution_L(Modint A[], int As, Modint B[], int Bs, Modint res[], int Rs,  Modint root = MD_PRIMITIVE_ROOT, void *mem = wmem){
  int i;
  int n;
  int n2;
  Modint*a;
  Modint*b;
  Modint r;
  n =max_L(As+Bs, Rs);
  for(n2=1;n2<n;n2*=2){
    ;
  }
  walloc1d(&a, n2, &mem);
  walloc1d(&b, n2, &mem);
  for(i=(0);i<(As);i++){
    a[i] = A[i];
  }
  int w54A8_wh = n2;
  for(i=(As);i<(w54A8_wh);i++){
    a[i].val = 0;
  }
  for(i=(0);i<(Bs);i++){
    b[i] = B[i];
  }
  int Zs_a1Ftk = n2;
  for(i=(Bs);i<(Zs_a1Ftk);i++){
    b[i].val = 0;
  }
  fft_L(n2, a, root, mem);
  fft_L(n2, b, root, mem);
  for(i=(0);i<(n2);i++){
    a[i] *= b[i];
  }
  fftinv_L(n2, a, root, mem);
  r = Modint(n2).inverse();
  for(i=(0);i<(Rs);i++){
    res[i] = a[i] * r;
  }
}
void convolution_L(Modint A[], int As, Modint res[], int Rs, Modint root = MD_PRIMITIVE_ROOT, void *mem = wmem){
  int i;
  int n;
  int n2;
  Modint*a;
  Modint r;
  n =max_L(2*As, Rs);
  for(n2=1;n2<n;n2*=2){
    ;
  }
  walloc1d(&a, n2, &mem);
  for(i=(0);i<(As);i++){
    a[i] = A[i];
  }
  int Ly6DsuN0 = n2;
  for(i=(As);i<(Ly6DsuN0);i++){
    a[i].val = 0;
  }
  fft_L(n2, a, root, mem);
  for(i=(0);i<(n2);i++){
    a[i] *= a[i];
  }
  fftinv_L(n2, a, root, mem);
  r = Modint(n2).inverse();
  for(i=(0);i<(Rs);i++){
    res[i] = a[i]*r;
  }
}
int N;
int H[100000];
int sz;
int hist[100000+1];
int len[100000];
Modint*arr[100000];
Modint*nx;
Modint df[100000+10];
int main(){
  int i;
  wmem = memarr;
  int mx;
  int x;
  int y;
  Modint res = 0;
  LHeap<int> hp;
  Comb<Modint> c;
  df[0] = df[1] = 1;
  for(i=(2);i<(100000+10);i++){
    df[i] = i * df[i-2];
  }
  rd(N);
  {
    int Q5VJL1cS;
    for(Q5VJL1cS=(0);Q5VJL1cS<(2*N);Q5VJL1cS++){
      rd(H[Q5VJL1cS]);H[Q5VJL1cS] += (-1);
    }
  }
  for(i=(0);i<(2*N);i++){
    hist[H[i]]++;
  }
  rsortA_L(100000, hist);
  while(hist[sz] > 1){
    sz++;
  }
  if(hist[0] > N){
    wt_L(0);
    wt_L('\n');
    return 0;
  }
  if(sz == 0){
    wt_L(df[2*N-1]);
    wt_L('\n');
    return 0;
  }
  for(i=(0);i<(sz);i++){
    int j;
    len[i] = hist[i] / 2 + 1;
    walloc1d(&arr[i], len[i]);
    arr[i][0] = 1;
    for(j=(1);j<(len[i]);j++){
      arr[i][j] = c.C(hist[i], 2*j) * df[2*j-1];
    }
  }
  hp.walloc(sz);
  hp.init(sz);
  for(i=(0);i<(sz);i++){
    hp.change(i, len[i]);
  }
  while(hp.size > 1){
    x = hp.pop();
    y = hp.pop();
    walloc1d(&nx, len[x] + len[y] - 1);
    convolution_L(arr[x], len[x], arr[y], len[y], nx, len[x] + len[y] - 1);
    arr[x] = nx;
    len[x] = len[x] + len[y] - 1;
    hp.change(x, len[x] + len[y] - 1);
  }
  x = hp.pop();
  for(i=(0);i<(len[x]);i++){
    if(2*N-1-2*i >= -1){
      if(i%2==0){
        if(2*N-1-2*i==-1){
          res +=arr[x][i] *1;
        }
        else{
          res +=arr[x][i] *df[2*N-1-2*i];
        }
      }
      else{
        if(2*N-1-2*i==-1){
          res -=arr[x][i] *1;
        }
        else{
          res -=arr[x][i] *df[2*N-1-2*i];
        }
      }
    }
  }
  wt_L(res);
  wt_L('\n');
  return 0;
}
// cLay varsion 20200926-1

// --- original code ---
// //working_memory=200MB
// #define MD 998244353
// int N, H[1d5];
// int sz, hist[1d5+1];
// 
// int len[1d5]; Modint *arr[1d5], *nx;
// Modint df[1d5+10];
// 
// {
//   int mx, x, y;
//   Modint res = 0;
//   LHeap<int> hp;
//   Comb<Modint> c;
// 
//   df[0] = df[1] = 1;
//   rep(i,2,1d5+10) df[i] = i * df[i-2];
// 
//   rd(N,(H--)(2*N));
// //  N = 1d5 / 2; rep(i,2*N) H[i] = rand()%1d5;
//   rep(i,2*N) hist[H[i]]++;
//   rsortA(1d5, hist);
//   while(hist[sz] > 1) sz++;
// 
//   if(hist[0] > N) wt(0), return 0;
//   if(sz == 0) wt(df[2*N-1]), return 0;
// 
//   // puts("hoge");
//   // wt(sz);
// 
//   rep(i,sz){
// //    wt(i,hist[i],hist[i+1],sz);
//     len[i] = hist[i] / 2 + 1;
//     walloc1d(&arr[i], len[i]);
// //    puts("ok");
//     arr[i][0] = 1;
//     rep(j,1,len[i]) arr[i][j] = c.C(hist[i], 2*j) * df[2*j-1];
//   }
// 
//   // wt(sz);
//   // rep(i,sz) wt(i, len[i], arr[i][0], arr[i][1]);
// 
//   hp.walloc(sz);
//   hp.init(sz);
//   rep(i,sz) hp.change(i, len[i]);
//   // puts("hoge");
//   // wt(hp.size);
//   while(hp.size > 1){
//     x = hp.pop();
//     y = hp.pop();
//     walloc1d(&nx, len[x] + len[y] - 1);
//     convolution(arr[x], len[x], arr[y], len[y], nx, len[x] + len[y] - 1);
//     // wt(len[x], len[y], len[x] + len[y] - 1);
//     arr[x] = nx;
//     len[x] = len[x] + len[y] - 1;
//     hp.change(x, len[x] + len[y] - 1);
//   }
//   // puts("piyo");
// 
//   x = hp.pop();
// //  wt("arr",arr[x](len[x]));
//   rep(i,len[x]){
//     if(2*N-1-2*i >= -1){
//       res if[i%2==0, +=, -=] arr[x][i] * if[2*N-1-2*i==-1, 1, df[2*N-1-2*i]];
//     }
//   }
//   wt(res);
//   // puts("aa");
// }


Measured memory_bytes: 9564