/*
 * Copyright (c) 1994-2000 FUJITSU LABORATORIES LIMITED 
 * All rights reserved.
 * 
 * FUJITSU LABORATORIES LIMITED ("FLL") hereby grants you a limited,
 * non-exclusive and royalty-free license to use, copy, modify and
 * redistribute, solely for non-commercial and non-profit purposes, the
 * computer program, "Risa/Asir" ("SOFTWARE"), subject to the terms and
 * conditions of this Agreement. For the avoidance of doubt, you acquire
 * only a limited right to use the SOFTWARE hereunder, and FLL or any
 * third party developer retains all rights, including but not limited to
 * copyrights, in and to the SOFTWARE.
 * 
 * (1) FLL does not grant you a license in any way for commercial
 * purposes. You may use the SOFTWARE only for non-commercial and
 * non-profit purposes only, such as academic, research and internal
 * business use.
 * (2) The SOFTWARE is protected by the Copyright Law of Japan and
 * international copyright treaties. If you make copies of the SOFTWARE,
 * with or without modification, as permitted hereunder, you shall affix
 * to all such copies of the SOFTWARE the above copyright notice.
 * (3) An explicit reference to this SOFTWARE and its copyright owner
 * shall be made on your publication or presentation in any form of the
 * results obtained by use of the SOFTWARE.
 * (4) In the event that you modify the SOFTWARE, you shall notify FLL by
 * e-mail at risa-admin@sec.flab.fujitsu.co.jp of the detailed specification
 * for such modification or the source code of the modified part of the
 * SOFTWARE.
 * 
 * THE SOFTWARE IS PROVIDED AS IS WITHOUT ANY WARRANTY OF ANY KIND. FLL
 * MAKES ABSOLUTELY NO WARRANTIES, EXPRESSED, IMPLIED OR STATUTORY, AND
 * EXPRESSLY DISCLAIMS ANY IMPLIED WARRANTY OF MERCHANTABILITY, FITNESS
 * FOR A PARTICULAR PURPOSE OR NONINFRINGEMENT OF THIRD PARTIES'
 * RIGHTS. NO FLL DEALER, AGENT, EMPLOYEES IS AUTHORIZED TO MAKE ANY
 * MODIFICATIONS, EXTENSIONS, OR ADDITIONS TO THIS WARRANTY.
 * UNDER NO CIRCUMSTANCES AND UNDER NO LEGAL THEORY, TORT, CONTRACT,
 * OR OTHERWISE, SHALL FLL BE LIABLE TO YOU OR ANY OTHER PERSON FOR ANY
 * DIRECT, INDIRECT, SPECIAL, INCIDENTAL, PUNITIVE OR CONSEQUENTIAL
 * DAMAGES OF ANY CHARACTER, INCLUDING, WITHOUT LIMITATION, DAMAGES
 * ARISING OUT OF OR RELATING TO THE SOFTWARE OR THIS AGREEMENT, DAMAGES
 * FOR LOSS OF GOODWILL, WORK STOPPAGE, OR LOSS OF DATA, OR FOR ANY
 * DAMAGES, EVEN IF FLL SHALL HAVE BEEN INFORMED OF THE POSSIBILITY OF
 * SUCH DAMAGES, OR FOR ANY CLAIM BY ANY OTHER PARTY. EVEN IF A PART
 * OF THE SOFTWARE HAS BEEN DEVELOPED BY A THIRD PARTY, THE THIRD PARTY
 * DEVELOPER SHALL HAVE NO LIABILITY IN CONNECTION WITH THE USE,
 * PERFORMANCE OR NON-PERFORMANCE OF THE SOFTWARE.
 *
 * $OpenXM: OpenXM_contrib2/asir2018/engine/M.c,v 1.2 2018/10/01 05:49:06 noro Exp $
*/
#include "ca.h"
#include "base.h"

void addum(int mod,UM p1,UM p2,UM pr)
{
  register int *c1,*c2,*cr,i,dmax,dmin;
    
  if ( DEG(p1) == -1 ) {
    cpyum(p2,pr);
    return;
  }
  if ( DEG(p2) == -1 ) {
    cpyum(p1,pr);
    return;
  }
  if ( DEG(p1) >= DEG(p2) ) {
    c1 = COEF(p1); c2 = COEF(p2); dmax = DEG(p1); dmin = DEG(p2);
  } else {
    c1 = COEF(p2); c2 = COEF(p1); dmax = DEG(p2); dmin = DEG(p1);
  }
  for ( i = 0, cr = COEF(pr); i <= dmin; i++ ) 
    cr[i] = ( c1[i] + c2[i] ) % mod;
  for ( ; i <= dmax; i++ ) 
    cr[i] = c1[i];
  if ( dmax == dmin ) 
    degum(pr,dmax);
  else 
    DEG(pr) = dmax;
}

void subum(int mod,UM p1,UM p2,UM pr)
{
  register int *c1,*c2,*cr,i;
  int dmax,dmin;

  if ( DEG(p1) == -1 ) {
    for ( i = DEG(pr) = DEG(p2), c2 = COEF(p2), cr = COEF(pr); 
        i >= 0; i-- ) 
      cr[i] = ( mod - c2[i] ) % mod;
    return;
  }
  if ( DEG(p2) == -1 ) {
    cpyum(p1,pr);
    return;
  }
  c1 = COEF(p1); c2 = COEF(p2); cr = COEF(pr);
  if ( DEG(p1) >= DEG(p2) ) { 
    dmax = DEG(p1); dmin = DEG(p2);
    for ( i = 0; i <= dmin; i++ ) 
      cr[i] = ( c1[i] + mod - c2[i] ) % mod;
    for ( ; i <= dmax; i++ ) 
      cr[i] = c1[i];
  } else {
    dmax = DEG(p2); dmin = DEG(p1);
    for ( i = 0; i <= dmin; i++ ) 
      cr[i] = ( c1[i] + mod - c2[i] ) % mod;
    for ( ; i <= dmax; i++ ) 
      cr[i] = ( mod - c2[i] ) % mod;
  }
  if ( dmax == dmin ) 
    degum(pr,dmax);
  else 
    DEG(pr) = dmax;
}
    
void pwrum(int mod,UM p,int e,UM pr)
{
  UM wt,ws;

  if ( e == 0 ) {
    DEG(pr) = 0; COEF(pr)[0] = 1;
  } else if ( DEG(p) < 0 ) 
    DEG(pr) = -1;
  else if ( e == 1 ) 
    cpyum(p,pr);
  else if ( DEG(p) == 0 ) {
    DEG(pr) = 0; COEF(pr)[0] = pwrm(mod,COEF(p)[0],e);
  } else {
    wt = W_UMALLOC(DEG(p)*e); ws = W_UMALLOC(DEG(p)*e);
    pwrum(mod,p,e/2,wt);
    if ( !(e%2) ) 
      mulum(mod,wt,wt,pr);
    else {
      mulum(mod,wt,wt,ws); mulum(mod,ws,p,pr);
    }
  }
}

void gcdum(int mod,UM p1,UM p2,UM pr)
{
  register int inv;
  UM t1,t2,q,tum;
  int drem;

  if ( DEG(p1) < 0 )
    cpyum(p2,pr);
  else if ( DEG(p2) < 0 )
    cpyum(p1,pr);
  else {
    if ( DEG(p1) >= DEG(p2) ) {
      t1 = p1; t2 = p2;
    } else {
      t1 = p2; t2 = p1;
    }
    q = W_UMALLOC(DEG(t1));
    while ( ( drem = divum(mod,t1,t2,q) ) >= 0 ) {
      tum = t1; t1 = t2; t2 = tum; DEG(t2) = drem;
    }
    inv = invm(COEF(t2)[DEG(t2)],mod);
    mulsum(mod,t2,inv,pr);
  }
}

void eucum(int mod,UM f1,UM f2,UM a,UM b)
{
  UM g1,g2,a1,a2,a3,wm,q,tum;
  int d,dr;

  d = DEG(f1) + DEG(f2) + 10;
  g1 = W_UMALLOC(d); g2 = W_UMALLOC(d); a1 = W_UMALLOC(d);
  a2 = W_UMALLOC(d); a3 = W_UMALLOC(d); wm = W_UMALLOC(d);
  q = W_UMALLOC(d);
  DEG(a1) = 0; COEF(a1)[0] = 1; DEG(a2) = -1;
  cpyum(f1,g1); cpyum(f2,g2);
  while ( 1 ) {
    dr = divum(mod,g1,g2,q); tum = g1; g1 = g2; g2 = tum;
    if ( ( DEG(g2) = dr ) == -1 ) 
      break;
    mulum(mod,a2,q,wm); subum(mod,a1,wm,a3); dr = divum(mod,a3,f2,q);
    tum = a1; a1 = a2; a2 = a3; a3 = tum; DEG(a3) = dr;
  }
  if ( COEF(g1)[0] != 1 )
    mulsum(mod,a2,invm(COEF(g1)[0],mod),a);
  else 
    cpyum(a2,a);
  mulum(mod,a,f1,wm);
  if ( DEG(wm) >= 0 ) 
    COEF(wm)[0] = ( COEF(wm)[0] + mod - 1 ) % mod;
  else {
    DEG(wm) = 0; COEF(wm)[0] = mod - 1;
  }
  divum(mod,wm,f2,q); mulsum(mod,q,mod-1,b);
#if 0
  t1 = W_UMALLOC(d);
  t2 = W_UMALLOC(d);
  t3 = W_UMALLOC(d);
  mulum(mod,a,f1,t1);
  mulum(mod,b,f2,t2);
  addum(mod,t1,t2,t3);
#endif
}

void eucum2(int mod,UM f1,UM f2,UM a,UM b)
{
  UM gk,gk1,gk2,ak,ak1,ak2,bk,bk1,bk2,q,t,wm1,wm2,wz;
  int d,inv;
  UM t1,t2;

  d = 2*(DEG(f1) + DEG(f2));
  gk = W_UMALLOC(d); gk1 = W_UMALLOC(d); gk2 = W_UMALLOC(d);
  ak = W_UMALLOC(d); ak1 = W_UMALLOC(d); ak2 = W_UMALLOC(d);
  bk = W_UMALLOC(d); bk1 = W_UMALLOC(d); bk2 = W_UMALLOC(d);
  q = W_UMALLOC(d); wm1 = W_UMALLOC(d); wm2 = W_UMALLOC(d);
  wz = W_UMALLOC(d);
  
  t1 = UMALLOC(1000);
  t2 = UMALLOC(1000);
  cpyum(f1,t1);
  cpyum(f2,t2);

  DEG(ak) = 0; COEF(ak)[0] = 1;
  DEG(ak1) = -1;
  DEG(bk) = -1;
  DEG(bk1) = 0; COEF(bk1)[0] = 1;

  cpyum(f1,gk); cpyum(f2,gk1);

  while ( 1 ) {
    /* ak*f1+bk*f2 = gk, ak1*f1+bk1*f2 = gk1 */
    cpyum(gk,gk2);
    DEG(gk2) = divum(mod,gk2,gk1,q);
    /* gk2 = gk - q*gk1 */
    if ( DEG(gk2) == -1 ) 
      break;
    /* ak2 = ak - q*ak1, bk2 = bk - q*bk1 */
    mulum(mod,ak1,q,wm1); subum(mod,ak,wm1,ak2);
    mulum(mod,bk1,q,wm1); subum(mod,bk,wm1,bk2);

    /* shift */
    t = ak; ak = ak1; ak1 = ak2; ak2 = t;
    t = bk; bk = bk1; bk1 = bk2; bk2 = t;
    t = gk; gk = gk1; gk1 = gk2; gk2 = t;
  }
  /* ak1*f1+bk1*f2 = gk1 = GCD(f1,f2) */
  mulum(mod,ak1,t1,wm1); 
  mulum(mod,bk1,t2,wm2);
  addum(mod,wm1,wm2,wz);
  if ( DEG(wz) != 0 )
    error("euc 1");

  DEG(ak1) = divum(mod,ak1,f2,q);
  DEG(bk1) = divum(mod,bk1,f1,q);
  mulum(mod,ak1,f1,wm1); 
  mulum(mod,bk1,f2,wm2);
  addum(mod,wm1,wm2,wz);
  if ( DEG(wz) != 0 )
    error("euc 2");


  if ( COEF(wz)[0] != 1 ) {
    inv = invm(COEF(wz)[0],mod);
    mulsum(mod,ak1,inv,a);
    mulsum(mod,bk1,inv,b);
  } else {
    cpyum(ak1,a);
    cpyum(bk1,b);
  }
}

void sqfrum(int index,int count,P f,int *nindex,struct oDUM **dcr,ML *pl)
{
  int i,j,m,n,d,dt,mod;
  UM wf,wdf,ws,wt,wgcd,mf,mgcd;
  UM *l;
  struct oDUM *dc;
  ML tp;

  n = UDEG(f);
  wf = W_UMALLOC(n);
  wdf = W_UMALLOC(n);
  ws = W_UMALLOC(n);
  wt = W_UMALLOC(n);
  wgcd = W_UMALLOC(n);

  mf = W_UMALLOC(n);
  mgcd = W_UMALLOC(n);

  for ( j = 0, d = n; j < count && d; ) {
    m = get_lprime(index++);
    if ( remqi(((Q)COEF(DC(f))),m) == 0 ) continue;

    ptoum(m,f,wf);
    diffum(m,wf,wdf);
    cpyum(wf,wt); cpyum(wdf,ws);
    gcdum(m,wt,ws,wgcd);
    dt = DEG(wgcd);

    if ( dt < d ) {
      d = dt;
      mod = m;
      cpyum(wf,mf); cpyum(wgcd,mgcd);
    } 
    j++;
  }
  *nindex = index;

  sqfrummain(mod,mf,mgcd,&dc);
  *dcr = dc;

  for ( n = 0; dc[n].f; n++ );
  *pl = tp = MLALLOC(n+1);
  tp->n = n;
  tp->mod = mod;

  for ( i = 0, l = (UM *)COEF(tp); dc[i].f; i++ ) {
    l[i] = UMALLOC(DEG(dc[i].f)*dc[i].n);
    pwrum(mod,dc[i].f,dc[i].n,l[i]);
  }
  l[i] = 0;
}

void sqfrummain(int mod,UM p,UM gcd,struct oDUM **dcp)
{
  int i,j,n;
  UM wp,wdp,wc,wd,ws,wt,wq;
  struct oDUM *dc;
  UM *f;

  i = DEG(p);

  wp = W_UMALLOC(i);
  wdp = W_UMALLOC(i);
  wt = W_UMALLOC(i);
  ws = W_UMALLOC(i);
  wc = W_UMALLOC(i);
  wd = W_UMALLOC(i);
  wq =  W_UMALLOC(i);

  f = (UM *) ALLOCA((i+2)*sizeof(UM));

  cpyum(p,wp);
  diffum(mod,wp,wdp);

  cpyum(wp,wt);
  divum(mod,wt,gcd,wc);

  cpyum(wdp,wt);
  divum(mod,wt,gcd,ws);

  diffum(mod,wc,wt);
  subum(mod,ws,wt,wd);

  for ( i = 1; DEG(wd) >= 0; i++ ) {
    cpyum(wc,ws); cpyum(wd,wt);
    gcdum(mod,ws,wt,wq);
    if ( DEG(wq) > 0 ) {
      f[i] = UMALLOC(DEG(wq));
      cpyum(wq,f[i]);

      cpyum(wc,ws);
      divum(mod,ws,f[i],wc);
      divum(mod,wd,f[i],ws);
      diffum(mod,wc,wt);
      subum(mod,ws,wt,wd);
    } else {
      f[i] = 0;
      cpyum(wd,ws);
      diffum(mod,wc,wt);
      subum(mod,ws,wt,wd);
    }
  
  }

  if ( DEG(wc) > 0 ) {
    DEG(wq) = 0;
    COEF(wq)[0] = invm(COEF(wc)[DEG(wc)],mod);
    f[i] = UMALLOC(DEG(wc));
    mulum(mod,wc,wq,f[i]);
    f[i+1] = 0;
    n = i + 1;
  } else {
    f[i] = 0;
    n = i;
  }
  
  for ( i = 1, j = 0; i < n; i++ )
    if ( f[i] ) j++;

  *dcp = dc = (struct oDUM *) CALLOC(j+1,sizeof(struct oDUM));

  for ( i = 1, j = 0; i < n; i++ ) 
    if ( f[i] ) {
      dc[j].n = i;
      dc[j].f = f[i];
      j++;
    }
  dc[j].n = 0;
  dc[j].f = 0;
}

void cpyum(UM p1,UM p2)
{
  register int *c1,*c2,i;

  for ( i = DEG(p2) = DEG(p1), c1 = COEF(p1), c2 = COEF(p2); 
      i >= 0; i-- ) 
    c2[i] = c1[i];
}

void clearum(UM p,int n)
{
  DEG(p) = -1;
  bzero(COEF(p),(n+1)*sizeof(int));
}

void degum(UM f,int n)
{
  register int i,*c;

  for ( i = n, c = COEF(f); ( i >= 0 ) && ( c[i] == 0 ); i-- );
  DEG(f) = i;
}

int deg(V v,P p)
{
  if ( !p )
    return ( -1 );
  else if ( NUM(p) )
    return ( 0 );
  else if ( VR(p) != v ) 
    return ( 0 );
  else if ( !smallz(DEG(DC(p))) ) {
    error("degree too large");
    return ( -1 );
  } else 
    return ( UDEG(p) );
}

LUM LUMALLOC(int n,int bound)
{
  LUM p;
  int **c;
  int i;

  p = (LUM)MALLOC(TRUESIZE(oLUM,n,int *));
  DEG(p) = n;
  for ( i = 0, c = (int **)COEF(p); i <= n; i++ ) {
    c[i] = (int *)MALLOC_ATOMIC((bound+1)*sizeof(int));
    bzero((char *)c[i],(bound+1)*sizeof(int));
  }
  return p;
}

/* dx = deg in x, dy = deg in y, c[i] <-> the coef of y^i (poly in x) */

BM BMALLOC(int dx,int dy)
{
  BM p;
  UM *c;
  int i;

  p = (BM)MALLOC(TRUESIZE(oBM,dy,UM));
  DEG(p) = dy;
  for ( i = 0, c = (UM *)COEF(p); i <= dy; i++ ) {
    c[i] = UMALLOC(dx);
    clearum(c[i],dx);
  }
  return p;
}

void mullum(int mod,int n,LUM f1,LUM f2,LUM fr)
{
  int max;
  int i,j;
  int **p1,**p2,*px;
  int *w,*w1,*w2;

  p1 = (int **)COEF(f1); p2 = (int **)COEF(f2);
  w = W_ALLOC(2*(n+1)); w1 = W_ALLOC(DEG(f1)); w2 = W_ALLOC(DEG(f2));
  for ( i = DEG(f1); i >= 0; i-- ) {
    for ( j = n - 1, px = p1[i]; ( j >= 0 ) && ( px[j] == 0 ); j-- );
    w1[i] = ( j == -1 ? 0 : 1 );
  }
  for ( i = DEG(f2); i >= 0; i-- ) {
    for ( j = n - 1, px = p2[i]; ( j >= 0 ) && ( px[j] == 0 ); j-- );
    w2[i] = ( j == -1 ? 0 : 1 );
  }
  for ( j = DEG(fr) = DEG(f1) + DEG(f2); j >= 0; j-- ) {
    for ( i = n - 1, px = COEF(fr)[j]; i >= 0; i-- ) 
      px[i] = 0;
    for ( max = MIN(DEG(f1),j), i = MAX(0,j-DEG(f2)); i <= max; i++ ) 
      if ( w1[i] != 0 && w2[j - i] != 0 ) {
        mulpadic(mod,n,(unsigned int *)p1[i],(unsigned int *)p2[j - i],(unsigned int *)w);
        addpadic(mod,n,(unsigned int *)w,(unsigned int *)px);
      }
  }
}

void cpylum(int bound,LUM p,LUM r)
{
  register int i,j;
  register int **pp,**ppr;

  DEG(r) = DEG(p);
  for ( i = 0, pp = COEF(p), ppr = COEF(r); 
    i <= DEG(p); i++ ) 
    for ( j = 0; j < bound; j++ ) 
      ppr[i][j] = pp[i][j];
}

int isequalum(UM f1,UM f2)
{
  int i;

  if ( DEG(f1) < 0 )
    if ( DEG(f2) < 0 )
      return 1;
    else
      return 0;
  else if ( DEG(f2) < 0 )
    return 0;
  else {
    if ( DEG(f1) != DEG(f2) )
      return 0;
    for ( i = 0; i <= DEG(f1); i++ )
      if ( COEF(f1)[i] != COEF(f2)[i] )
        break;
    if ( i < DEG(f1) )
      return 0;
    else
      return 1;
  }
}

void pwrlum(int mod,int bound,LUM p,int n,LUM r)
{
  LUM t,s;

  if ( n == 0 ) {
    DEG(r) = 0;
    COEF(r)[0][0] = 1;
  } else if ( DEG(p) < 0 )
    DEG(r) = -1;
  else if ( n == 1 ) 
    cpylum(bound,p,r);
  else {
    W_LUMALLOC(DEG(p)*n,bound,t);
    pwrlum(mod,bound,p,n/2,t);
    if ( !(n%2) ) 
      mullum(mod,bound,t,t,r);
    else {
      W_LUMALLOC(DEG(p)*n,bound,s);
      mullum(mod,bound,t,t,s);
      mullum(mod,bound,s,p,r);
    }
  }
}

int **almat(int n,int m)
{
  int **mat,i;

  mat = (int **)MALLOC(n*sizeof(int *));
  for ( i = 0; i < n; i++ )
    mat[i] = (int *)CALLOC(m,sizeof(int));
  return mat;
}

#if defined(__GNUC__) && SIZEOF_LONG == 8
mp_limb_t **almat64(int n,int m)
{
  mp_limb_t **mat,i;

  mat = (mp_limb_t **)MALLOC(n*sizeof(mp_limb_t *));
  for ( i = 0; i < n; i++ )
    mat[i] = (mp_limb_t *)CALLOC(m,sizeof(mp_limb_t));
  return mat;
}
#endif

void mini(int mod,UM f,UM fr)
{
  register int i,j,**c,*ptr;
  int d,dr,dm,n;
  UM w,q;

  n = DEG(f); c = (int **)ALLOCA(n*sizeof(int *));
  for ( i = 0; i < n; i++ ) {
    c[i] = (int *)ALLOCA(n*sizeof(int));
    bzero((char *)c[i],(int)(n*sizeof(int)));
  }
  w = W_UMALLOC( mod + n + 10 ); q = W_UMALLOC( mod + n + 10 );
  for ( i = 1; ( d = ( mod * i ) ) < n; i++ ) c[d][i - 1] = 1;
  DEG(w) = d;
  for ( j = 0; j < d; j++ )
    COEF(w)[j] = 0;
  COEF(w)[d] = 1;
  for ( ; (i < n) && ((dr = divum(mod,w,f,q)) >= 0); i++ ) {
    for ( j = dr; j >= 0; j-- ) 
      COEF(w)[j + mod] = c[j][i - 1] = COEF(w)[j];
    for ( j = mod - 1; j >= 0; j-- ) 
      COEF(w)[j] = 0;
    DEG(w) = dr + mod;
  }
  for ( i = 1; i < n; i++ ) 
    c[i][i - 1] = ( c[i][i - 1] + mod - 1 ) % mod;
  if ( ( dm = minimain(mod,n,n - 1,c) ) != -1 ) 
    for ( i = 0, ptr = COEF(fr), ptr[0] = 0; i <= dm; i++ ) 
      ptr[i + 1] = c[0][i];
  else 
    COEF(fr)[0] = 1;
  DEG(fr) = dm + 1;
}

int minimain(int mod,int n,int m,int **c)
{
  register int *ptr,*ci,*p;
  register int i,l,a,j,b,inv;
  int *tmp;

  for ( j = 0; j < m; j++ ) {
    for ( i = j; (n > i) && !c[i][j]; i++ );
    if ( i == n ) {
      for ( i = j, j = j - 1; j >= 0; j-- ) 
        c[0][j] = c[j][i];
      c[0][i] = mod - 1;
      return( i );
    }
    if ( i != j ) {
      tmp = c[i]; c[i] = c[j]; c[j] = tmp;
    }
    ptr = c[j]; inv = invm((ptr[j] + mod) % mod,mod);
    for ( l = j, p = ptr+l; l < m; l++ ) { 
      a = (*p * inv) % mod;
      *p++ = (a<0?a+mod:a);
    }
    for ( i = 0; i < n; i++ )
      if ( (a = -c[i][j]) && (i != j) ) {
        for ( l = j+1, p = ptr+l, ci = c[i]+l; l < m; l++ ) {
          b = (*p++ * a + *ci) % mod;
          *ci++ = (b<0?b+mod:b);
        }
        c[i][j] = 0;
      }
  }
  return (-1);
}

#if defined(__GNUC__)
const
#endif
int sprime[] = {
  2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,
  53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,
  127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,
  199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,
  283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,
  383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,
  467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,
  577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,
  661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,
  769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,
  877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,
  983,991,997,1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,
  1087,1091,1093,1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,
  1193,1201,1213,1217,1223,1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,
  1297,1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,
  1429,1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,1499,1511,
  1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607,1609,1613,
  1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709,1721,1723,1733,
  1741,1747,1753,1759,1777,1783,1787,1789,1801,1811,1823,1831,1847,1861,1867,
  1871,1873,1877,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,1979,1987,
  1993,1997,1999,2003,2011,2017,2027,2029,2039,2053,2063,2069,2081,2083,2087,
  2089,2099,2111,2113,2129,2131,2137,2141,2143,2153,2161,2179,2203,2207,2213,
  2221,2237,2239,2243,2251,2267,2269,2273,2281,2287,2293,2297,2309,2311,2333,
  2339,2341,2347,2351,2357,2371,2377,2381,2383,2389,2393,2399,2411,2417,2423,
  2437,2441,2447,2459,2467,2473,2477,2503,2521,2531,2539,2543,2549,2551,2557,
  2579,2591,2593,2609,2617,2621,2633,2647,2657,2659,2663,2671,2677,2683,2687,
  2689,2693,2699,2707,2711,2713,2719,2729,2731,2741,2749,2753,2767,2777,2789,
  2791,2797,2801,2803,2819,2833,2837,2843,2851,2857,2861,2879,2887,2897,2903,
  2909,2917,2927,2939,2953,2957,2963,2969,2971,2999,3001,3011,3019,3023,3037,
  3041,3049,3061,3067,3079,3083,3089,3109,3119,3121,3137,3163,3167,3169,3181,
  3187,3191,3203,3209,3217,3221,3229,3251,3253,3257,3259,3271,3299,3301,3307,
  3313,3319,3323,3329,3331,3343,3347,3359,3361,3371,3373,3389,3391,3407,3413,
  3433,3449,3457,3461,3463,3467,3469,3491,3499,3511,3517,3527,3529,3533,3539,
  3541,3547,3557,3559,3571,3581,3583,3593,3607,3613,3617,3623,3631,3637,3643,
  3659,3671,3673,3677,3691,3697,3701,3709,3719,3727,3733,3739,3761,3767,3769,
  3779,3793,3797,3803,3821,3823,3833,3847,3851,3853,3863,3877,3881,3889,3907,
  3911,3917,3919,3923,3929,3931,3943,3947,3967,3989,4001,4003,4007,4013,4019,
  4021,4027,4049,4051,4057,4073,4079,4091,4093,4099,4111,4127,4129,4133,4139,
  4153,4157,4159,4177,4201,4211,4217,4219,4229,4231,4241,4243,4253,4259,4261,
  4271,4273,4283,4289,4297,4327,4337,4339,4349,4357,4363,4373,4391,4397,4409,
  4421,4423,4441,4447,4451,4457,4463,4481,4483,4493,4507,4513,4517,4519,4523,
  4547,4549,4561,4567,4583,4591,4597,4603,4621,4637,4639,4643,4649,4651,4657,
  4663,4673,4679,4691,4703,4721,4723,4729,4733,4751,4759,4783,4787,4789,4793,
  4799,4801,4813,4817,4831,4861,4871,4877,4889,4903,4909,4919,4931,4933,4937,
  4943,4951,4957,4967,4969,4973,4987,4993,4999,5003,5009,5011,5021,5023,5039,
  5051,5059,5077,5081,5087,5099,5101,5107,5113,5119,5147,5153,5167,5171,5179,
  5189,5197,5209,5227,5231,5233,5237,5261,5273,5279,5281,5297,5303,5309,5323,
  5333,5347,5351,5381,5387,5393,5399,5407,5413,5417,5419,5431,5437,5441,5443,
  5449,5471,5477,5479,5483,5501,5503,5507,5519,5521,5527,5531,5557,5563,5569,
  5573,5581,5591,5623,5639,5641,5647,5651,5653,5657,5659,5669,5683,5689,5693,
  5701,5711,5717,5737,5741,5743,5749,5779,5783,5791,5801,5807,5813,5821,5827,
  5839,5843,5849,5851,5857,5861,5867,5869,5879,5881,5897,5903,5923,5927,5939,
  5953,5981,5987,6007,6011,6029,6037,6043,6047,6053,6067,6073,6079,6089,6091,
  6101,6113,6121,6131,6133,6143,6151,6163,6173,6197,6199,6203,6211,6217,6221,
  6229,6247,6257,6263,6269,6271,6277,6287,6299,6301,6311,6317,6323,6329,6337,
  6343,6353,6359,6361,6367,6373,6379,6389,6397,6421,6427,6449,6451,6469,6473,
  6481,6491,6521,6529,6547,6551,6553,6563,6569,6571,6577,6581,6599,6607,6619,
  6637,6653,6659,6661,6673,6679,6689,6691,6701,6703,6709,6719,6733,6737,6761,
  6763,6779,6781,6791,6793,6803,6823,6827,6829,6833,6841,6857,6863,6869,6871,
  6883,6899,6907,6911,6917,6947,6949,6959,6961,6967,6971,6977,6983,6991,6997,
  7001,7013,7019,7027,7039,7043,7057,7069,7079,7103,7109,7121,7127,7129,7151,
  7159,7177,7187,7193,7207,7211,7213,7219,7229,7237,7243,7247,7253,7283,7297,
  7307,7309,7321,7331,7333,7349,7351,7369,7393,7411,7417,7433,7451,7457,7459,
  7477,7481,7487,7489,7499,7507,7517,7523,7529,7537,7541,7547,7549,7559,7561,
  7573,7577,7583,7589,7591,7603,7607,7621,7639,7643,7649,7669,7673,7681,7687,
  7691,7699,7703,7717,7723,7727,7741,7753,7757,7759,7789,7793,7817,7823,7829,
  7841,7853,7867,7873,7877,7879,7883,7901,7907,7919,7927,7933,7937,7949,7951,
  7963,7993,8009,8011,8017,8039,8053,8059,8069,8081,8087,8089,8093,8101,8111,
  8117,8123,8147,8161,8167,8171,8179,8191,8209,8219,8221,8231,8233,8237,8243,
  8263,8269,8273,8287,8291,8293,8297,8311,8317,8329,8353,8363,8369,8377,8387,
  8389,8419,8423,8429,8431,8443,8447,8461,8467,8501,8513,8521,8527,8537,8539,
  8543,8563,8573,8581,8597,8599,8609,8623,8627,8629,8641,8647,8663,8669,8677,
  8681,8689,8693,8699,8707,8713,8719,8731,8737,8741,8747,8753,8761,8779,8783,
  8803,8807,8819,8821,8831,8837,8839,8849,8861,8863,8867,8887,8893,8923,8929,
  8933,8941,8951,8963,8969,8971,8999,9001,9007,9011,9013,9029,9041,9043,9049,
  9059,9067,9091,9103,9109,9127,9133,9137,9151,9157,9161,9173,9181,9187,9199,
  9203,9209,9221,9227,9239,9241,9257,9277,9281,9283,9293,9311,9319,9323,9337,
  9341,9343,9349,9371,9377,9391,9397,9403,9413,9419,9421,9431,9433,9437,9439,
  9461,9463,9467,9473,9479,9491,9497,9511,9521,9533,9539,9547,9551,9587,9601,
  9613,9619,9623,9629,9631,9643,9649,9661,9677,9679,9689,9697,9719,9721,9733,
  9739,9743,9749,9767,9769,9781,9787,9791,9803,9811,9817,9829,9833,9839,9851,
  9857,9859,9871,9883,9887,9901,9907,9923,9929,9931,9941,9949,9967,9973,10007, 
  10009,10037,10039,10061,10067,10069,10079,10091,10093,10099,
  10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,
  10181,10193,10211,10223,10243,10247,10253,10259,10267,10271,
  10273,10289,10301,10303,10313,10321,10331,10333,10337,10343,
  10357,10369,10391,10399,10427,10429,10433,10453,10457,10459,
  10463,10477,10487,10499,10501,10513,10529,10531,10559,10567,
  10589,10597,10601,10607,10613,10627,10631,10639,10651,10657,
  10663,10667,10687,10691,10709,10711,10723,10729,10733,10739,
  10753,10771,10781,10789,10799,10831,10837,10847,10853,10859,
  10861,10867,10883,10889,10891,10903,10909,10937,10939,10949,
  10957,10973,10979,10987,10993,11003,11027,11047,11057,11059,
  11069,11071,11083,11087,11093,11113,11117,11119,11131,11149,
  11159,11161,11171,11173,11177,11197,11213,11239,11243,11251,
  11257,11261,11273,11279,11287,11299,11311,11317,11321,11329,
  11351,11353,11369,11383,11393,11399,11411,11423,11437,11443,
  11447,11467,11471,11483,11489,11491,11497,11503,11519,11527,
  11549,11551,11579,11587,11593,11597,11617,11621,11633,11657,
  11677,11681,11689,11699,11701,11717,11719,11731,11743,11777,
  11779,11783,11789,11801,11807,11813,11821,11827,11831,11833,
  11839,11863,11867,11887,11897,11903,11909,11923,11927,11933,
  11939,11941,11953,11959,11969,11971,11981,11987,12007,12011,
  12037,12041,12043,12049,12071,12073,12097,12101,12107,12109,
  12113,12119,12143,12149,12157,12161,12163,12197,12203,12211,
  12227,12239,12241,12251,12253,12263,12269,12277,12281,12289,
  12301,12323,12329,12343,12347,12373,12377,12379,12391,12401,
  12409,12413,12421,12433,12437,12451,12457,12473,12479,12487,
  12491,12497,12503,12511,12517,12527,12539,12541,12547,12553,
  12569,12577,12583,12589,12601,12611,12613,12619,12637,12641,
  12647,12653,12659,12671,12689,12697,12703,12713,12721,12739,
  12743,12757,12763,12781,12791,12799,12809,12821,12823,12829,
  12841,12853,12889,12893,12899,12907,12911,12917,12919,12923,
  12941,12953,12959,12967,12973,12979,12983,13001,13003,13007,
  13009,13033,13037,13043,13049,13063,13093,13099,13103,13109,
  13121,13127,13147,13151,13159,13163,13171,13177,13183,13187,
  13217,13219,13229,13241,13249,13259,13267,13291,13297,13309,
  13313,13327,13331,13337,13339,13367,13381,13397,13399,13411,
  13417,13421,13441,13451,13457,13463,13469,13477,13487,13499,
  13513,13523,13537,13553,13567,13577,13591,13597,13613,13619,
  13627,13633,13649,13669,13679,13681,13687,13691,13693,13697,
  13709,13711,13721,13723,13729,13751,13757,13759,13763,13781,
  13789,13799,13807,13829,13831,13841,13859,13873,13877,13879,
  13883,13901,13903,13907,13913,13921,13931,13933,13963,13967,
  13997,13999,14009,14011,14029,14033,14051,14057,14071,14081,
  14083,14087,14107,14143,14149,14153,14159,14173,14177,14197,
  14207,14221,14243,14249,14251,14281,14293,14303,14321,14323,
  14327,14341,14347,14369,14387,14389,14401,14407,14411,14419,
  14423,14431,14437,14447,14449,14461,14479,14489,14503,14519,
  14533,14537,14543,14549,14551,14557,14561,14563,14591,14593,
  14621,14627,14629,14633,14639,14653,14657,14669,14683,14699,
  14713,14717,14723,14731,14737,14741,14747,14753,14759,14767,
  14771,14779,14783,14797,14813,14821,14827,14831,14843,14851,
  14867,14869,14879,14887,14891,14897,14923,14929,14939,14947,
  14951,14957,14969,14983,15013,15017,15031,15053,15061,15073,
  15077,15083,15091,15101,15107,15121,15131,15137,15139,15149,
  15161,15173,15187,15193,15199,15217,15227,15233,15241,15259,
  15263,15269,15271,15277,15287,15289,15299,15307,15313,15319,
  15329,15331,15349,15359,15361,15373,15377,15383,15391,15401,
  15413,15427,15439,15443,15451,15461,15467,15473,15493,15497,
  15511,15527,15541,15551,15559,15569,15581,15583,15601,15607,
  15619,15629,15641,15643,15647,15649,15661,15667,15671,15679,
  15683,15727,15731,15733,15737,15739,15749,15761,15767,15773,
  15787,15791,15797,15803,15809,15817,15823,15859,15877,15881,
  15887,15889,15901,15907,15913,15919,15923,15937,15959,15971,
  15973,15991,16001,16007,16033,16057,16061,16063,16067,16069,
  16073,16087,16091,16097,16103,16111,16127,16139,16141,16183,
  16187,16189,16193,16217,16223,16229,16231,16249,16253,16267,
  16273,16301,16319,16333,16339,16349,16361,16363,16369,16381,
  0
};