struct cpx{
  cpx(){}
  cpx(double aa):a(aa) {}
  cpx(double aa, double bb):a(aa),b(bb) {}
  double a;
  double b;
  double modsq(void) const {
    return a * a + b * b;
  }
  cpx bar(void) const {
    return cpx(a, -b);
  }
};
cpx operator +(cpx a, cpx b) {
  return cpx(a.a + b.a, a.b + b.b);
}
cpx operator *(cpx a, cpx b) {
  return cpx(a.a * b.a - a.b * b.b, a.a * b.b + a.b * b.a);
}
cpx operator /(cpx a, cpx b) {
  cpx r = a * b.bar();
  return cpx(r.a / b.modsq(), r.b / b.modsq());
}
cpx EXP(double theta) {
  return cpx(cos(theta),sin(theta));
}
const double two_pi = 4 * acos(0);
// in:     input array
// out:    output array
// step:   {SET TO 1} (used internally)
// size:   length of the input/output {MUST BE A POWER OF 2} (use nextPowerOf2)
// dir:    either plus or minus one (direction of the FFT)
// RESULT: out[k] = \sum_{j=0}^{size - 1} in[j] * exp(dir * 2pi * i * j * k / size)
void FFT(cpx *in, cpx *out, int step, int size, int dir) {
  if(size < 1) return;
  if(size == 1) {
    out[0] = in[0];
    return;
  }
  FFT(in, out, step * 2, size / 2, dir);
  FFT(in + step, out + size / 2, step * 2, size / 2, dir);
  for(int i = 0 ; i < size / 2 ; i++) {
    cpx even = out[i];
    cpx odd = out[i + size / 2];
    out[i] = even + EXP(dir * two_pi * i / size) * odd;
    out[i + size / 2] = even + EXP(dir * two_pi * (i + size / 2) / size) * odd;
  }
}
unsigned int nextPowerOf2(unsigned int n) {
    unsigned int p = 1;
    if (n && !(n & (n - 1))) return n;
    while (p < n) p <<= 1;
    return p;
}