You are given a positive integer $n$.

Find a sequence of fractions $\frac{a_i}{b_i}$, $i = 1 \ldots k$ (where $a_i$ and $b_i$ are positive integers) for some $k$ such that:

$$ \begin{cases} \text{$b_i$ divides $n$, $1 < b_i < n$ for $i = 1 \ldots k$} \\ \text{$1 \le a_i < b_i$ for $i = 1 \ldots k$} \\ \text{$\sum\limits_{i=1}^k \frac{a_i}{b_i} = 1 - \frac{1}{n}$} \end{cases} $$


-----Input-----

The input consists of a single integer $n$ ($2 \le n \le 10^9$).


-----Output-----

In the first line print "YES" if there exists such a sequence of fractions or "NO" otherwise.

If there exists such a sequence, next lines should contain a description of the sequence in the following format.

The second line should contain integer $k$ ($1 \le k \le 100\,000$) — the number of elements in the sequence. It is guaranteed that if such a sequence exists, then there exists a sequence of length at most $100\,000$.

Next $k$ lines should contain fractions of the sequence with two integers $a_i$ and $b_i$ on each line.


-----Examples-----
Input
2

Output
NO

Input
6

Output
YES
2
1 2
1 3



-----Note-----

In the second example there is a sequence $\frac{1}{2}, \frac{1}{3}$ such that $\frac{1}{2} + \frac{1}{3} = 1 - \frac{1}{6}$.

Difficulty: HARD
Tags: Mathematics

Starter code:

Solutions:
from math import gcd

def check(e, f, n):
	i = 1
	while i * e <= n:
		if (n - i * e) % f == 0:
			return (n - i * e) // f
		i += 1

def prime(x):
	b = []
	i = 2
	while i * i <= x:
		if x % i == 0:
			if x // i == i:
				b.append(i)
			else:
				b.extend([i, x // i])
		i += 1
	return b
k = int(input())
a = prime(k)
a.sort()
(e, f) = (-1, -1)
l = len(a)
for i in range(l):
	if gcd(a[i], a[l - 1 - i]) == 1:
		e = a[i]
		f = a[l - i - 1]
		break
if e == -1:
	print('NO')
else:
	x = check(e, f, k - 1)
	y = (k - 1 - x * f) // e
	print('YES')
	print(2)
	print(x, e)
	print(y, f)


def egcd(a, b):
	if a == 0:
		return (b, 0, 1)
	(gcd, x1, y1) = egcd(b % a, a)
	x = y1 - b // a * x1
	y = x1
	return (gcd, x, y)

def main():
	n = int(input())
	div = []
	d = set()
	m = n
	i = 2
	while i * i <= n:
		cnt = 0
		while n % i == 0:
			cnt += 1
			n //= i
			d.add(i)
		if cnt > 0:
			div.append((cnt, i))
		i += 1
	if n > 1:
		div.append((1, n))
		d.add(n)
	for i in d:
		if i == m:
			d.remove(i)
			break
	if len(d) < 2:
		print('NO')
		return
	ans1 = 1
	for i in range(div[0][0]):
		ans1 *= div[0][1]
	ans2 = 1
	for i in div[1:]:
		for j in range(i[0]):
			ans2 *= i[1]
	(gcd, x, y) = egcd(ans2, -ans1)
	if x < 0 or y < 0:
		(gcd, x, y) = egcd(ans1, -ans2)
		(x, y) = (y, x)
	print('YES')
	print(2)
	if ans2 * x + ans1 * y == m - 1:
		print(x, ans1)
		print(y, ans2)
	elif ans2 * (ans1 - x) + ans1 * y == m - 1:
		print(ans1 - x, ans1)
		print(y, ans2)
	elif ans2 * x + ans1 * (ans2 - y) == m - 1:
		print(x, ans1)
		print(ans2 - y, ans2)
	else:
		print(ans1 - x, ans1)
		print(ans2 - y, ans2)
	return

def __starting_point():
	main()
__starting_point()


import math

def ps(n):
	a = []
	i = 1
	while i <= math.sqrt(n):
		if n % i == 0:
			if n / i == i:
				a.append(i)
			else:
				a.extend([i, n // i])
		i = i + 1
	return a

def gcd(a, b):
	if a % b == 0:
		return b
	else:
		return gcd(b, a % b)

def sol(a, b, n):
	i = 0
	while i * a <= n:
		if (n - i * a) % b == 0:
			return int((n - i * a) / b)
		i = i + 1
n = int(input())
a = ps(n)
a.sort()
e = -1
d = len(a)
for i in range(1, (len(a) + 1) // 2):
	if gcd(a[i], a[d - 1 - i]) == 1:
		e = a[i]
		f = a[d - i - 1]
if e == -1:
	print('NO')
else:
	a = sol(e, f, n - 1)
	b = (n - 1 - a * f) // e
	print('YES')
	print(2)
	print(a, e)
	print(b, f)


from math import sqrt

def phi(u):
	ans = u
	for i in range(2, int(sqrt(n)) + 1):
		if u % i == 0:
			while u % i == 0:
				u = u / i
			ans = ans - int(ans / i)
	if n > 1:
		ans = ans - int(ans / n)
	return ans

def binpow(u, a, mod):
	ans = 1
	if a == 0:
		return 1
	while a > 0:
		if a % 2 == 0:
			u = u ** 2 % mod
			a = int(a / 2)
		else:
			ans = ans * u % mod
			a = a - 1
	return int(ans)
n = int(input())
b1 = 1
b2 = 0
nn = n
for i in range(2, int(sqrt(n)) + 1):
	if n % i == 0:
		while nn % i == 0:
			b1 = b1 * i
			nn = nn / i
		b2 = int(n / b1)
		break
if b2 < 2:
	print('NO')
	exit()
a1 = b1 - binpow(b2, phi(b1) - 1, b1)
a2 = b2 - int((a1 * b2 + 1) / b1)
print('YES')
print(2)
print(a1, b1)
print(a2, b2)


from math import sqrt
from itertools import count, islice
from fractions import Fraction

def isPrime(n):
	return n > 1 and all((n % i for i in islice(count(2), int(sqrt(n) - 1))))

def factors(n: int):
	_factors = []
	for i in range(2, int(sqrt(n)) + 1):
		times = 0
		while n % i == 0:
			times += 1
			n //= i
		if times:
			_factors.append(i ** times)
	if n > 1:
		_factors.append(n)
	return _factors
n = int(input())
_f = factors(n)
sz = len(_f)
if sz < 2:
	print('NO')
else:
	print('YES\n2')
	for i in range(1, _f[0]):
		(num, den) = (i, _f[0])
		frac = Fraction(num, den)
		frac = Fraction(n - 1, n) - frac
		if frac.denominator < n:
			print(num, den)
			print(frac.numerator, frac.denominator)
			break