from fractions import Fraction

def div_row(l, row, m):
	for i in range(len(l[row])): l[row][i] = l[row][i] / m


def add_row(l, row1, row2, m=1):
	for i in range(len(l[row1])): l[row1][i] = l[row1][i] + l[row2][i] * m


def process(l):
	# Converts matrix to fractions, and gets terminal states.
	# Transforms L into (I - L^T) | (1,0,0,...)^T
	new_l = [[0]*len(l) for x in range(len(l))]
	terminal = []
	for i in range(len(l)):
		d = sum(l[i])
		if d == 0: terminal.append(i)
		for j in range(len(l)):
			new_l[j][i] = -Fraction(l[i][j], 1 if d == 0 else d)
			if i == j: new_l[i][j] = 1 + new_l[i][j]
	
	for i in range(len(l)): new_l[i].append(Fraction(i == 0))
	return (new_l, terminal)


def gauss_elim(l):
	for i in range(len(l)):	# Column
		for j in range(len(l)):	# Row
			if i == j:
				if l[j][i] == 1: continue
				div_row(l, j, l[j][i])
			else:
				if l[j][i] == 0: continue
				add_row(l, j, i, Fraction(-l[j][i], l[i][i]))


def gcd(a, b):
	while b > 0:
		a, b = b, a % b
	return a

def lcm(a, b):
	return a * b / gcd(a, b)


def simplify(p):
	denom = [x.denominator for x in p]
	curr_lcm = denom[0]
	for i in range(1, len(denom)):
		curr_lcm = lcm(curr_lcm, denom[i])

	for i in range(len(p)):
		p[i] = p[i].numerator * curr_lcm / denom[i] 

	p.append(curr_lcm)


def solution(l):
	# L^n[i][j] = Probability of arriving at state j starting from state i in n steps
	# Sum(L^n) from 0 to n = (I - L)^{-1}.
	# Since we only want the first row, we can do some simplifications by
	# row reducing with the transpose, and augment with the (1,0,0,0, ...)^T
	l, terminal = process(l)
	print(l)
	gauss_elim(l)

	probs = [x[-1] for i,x in enumerate(l) if i in terminal]
	simplify(probs)
	return probs