import math
import cmath


def dft(p):
	"""Quadratic formula from the definition."""
	N = len(p)
	w = cmath.exp(-2*math.pi*1j/N)  # primitive N-th square root of 1
	x = [0]*N
	y = [0]*N
	for k in range(N):
		x[k] = w**k
		for n in range(N):
			y[k] += p[n] * x[k]**n
	return y


def compute_inverse(N1, N2):
	for i in range(N2):
		if N1*i % N2 == 1:
			return i
	raise ValueError("Failed to find an inverse to {0} mod {1}.".format(N1, N2))


def prime_fft(p, divisors):
	"""https://en.wikipedia.org/wiki/Prime-factor_FFT_algorithm"""
	if len(divisors) == 1:
		return dft(p)
	N = len(p)
	N1 = divisors[0]
	N2 = N//N1
	N1_inv = compute_inverse(N1, N2)
	N2_inv = compute_inverse(N2, N1)

	ys = []
	for n1 in range(N1):  # compute rows
		p_ = [p[(n2*N1+n1*N2) % N] for n2 in range(N2)]
		ys.append(prime_fft(p_, divisors[1:]))

	for k2 in range(N2):  # compute cols
		p_ = [row[k2] for row in ys]
		y_ = dft(p_)
		for (yi, row) in zip(y_, ys):  # update col
			row[k2] = yi

	# remap and output
	res = [0]*N
	for k1 in range(N1):
		for k2 in range(N2):
			res[(k1*N2*N2_inv+k2*N1*N1_inv) % N] = ys[k1][k2]
	return res


# arr = [9,8,5,3,8,4,9,7,0,9,5,6,6,2,4]
# print(dft(arr))
# print()
# print(prime_fft(arr,[3,5]))

"""
def fft(x):
    N = len(x)
    if N <= 1: return x
    even = fft(x[0::2])
    odd = fft(x[1::2])
    T = [cmath.exp(-2j*math.pi*k/N)*odd[k] for k in range(N//2)]
    return [even[k] + T[k] for k in range(N//2)] + \
           [even[k] - T[k] for k in range(N//2)]


def fft_mixed(x, r1, r2):
	A = dft(x)
	for k0 in range(0, r2):
		pass


def bit_reverse(x, width):
	y = 0
	for i in range(width):
		y <<= 1
		y |= x&1
		x >>= 1
	return y


def bit_reverse_seq(x):
	n = len(x)
	width = round(math.log2(n))
	return [x[bit_reverse(i, width)] for i in range(n)]


def fft2(x):
	y = bit_reverse_seq(x)
	n = len(x)
	omegas = [(-2*math.pi*1j/n)**i for i in range(0, n)]
	b = 1
	while b<n:
		for j in range(0, n, 2*b):
			for k in range(0, b):
				alpha = omegas[n*k//(2*b)]
				u = y[j+k]
				t = alpha*y[j+k+b]
				y[j+k] = u+t
				y[j+k+b] = u-t
		b *= 2
	return y
"""