



import math

def 			A_      (	__a     :				int		):

				"""simple docstring"""

				if 1 < number < 4:
								# 2 and 3 are primes
								return True
				elif number < 2 or number % 2 == 0 or number % 3 == 0:
								# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
								return False

				# All primes number are in format of 6k +/- 1
				for i in range(5 ,		int(math.sqrt(a_		) + 1		) ,		6		):
								if number % i == 0 or number % (i + 2) == 0:
												return False
				return True

def 			A_      (	__a     :				float = 0.1		):

				"""simple docstring"""
				a__				      =		3
				a__				      =		3

				while primes / (2 * j - 1) >= ratio:
								for i in range(j * j + j + 1 ,		(j + 2) * (j + 2) ,		j + 1		):
												primes += is_prime(a_		)
								j += 2
				return j


if __name__ == "__main__":
	import doctest

	doctest.testmod()






def 			A_      (	__a     :				Any		):

				"""simple docstring"""
				if not head:
								return True
				# split the list to two parts
				a__      ,			a__				      =		head.next, head
				while fast and fast.next:
								a__				      =		fast.next.next
								a__				      =		slow.next
				a__				      =		slow.next
				a__				      =		None  # Don't forget here! But forget still works!
				# reverse the second part
				a__				      =		None
				while second:
								a__				      =		second.next
								a__				      =		node
								a__				      =		second
								a__				      =		nxt
				# compare two parts
				# second part has the same or one less node
				while node:
								if node.val != head.val:
												return False
								a__				      =		node.next
								a__				      =		head.next
				return True

def 			A_      (	__a     :				int		):

				"""simple docstring"""
				if not head or not head.next:
								return True

				# 1. Get the midpoint (slow)
				a__				      =		a__				      =		a__				      =		head
				while fast and fast.next:
								a__      ,			a__				      =		fast.next.next, slow.next

				# 2. Push the second half into the stack
				a__				      =		[slow.val]
				while slow.next:
								a__				      =		slow.next
								stack.append(slow.val		)

				# 3. Comparison
				while stack:
								if stack.pop() != cur.val:
												return False
								a__				      =		cur.next

				return True




def 			A_      (	__a     :				Any		):

				"""simple docstring"""
				if not head or not head.next:
								return True
				a__				      =		{}
				a__				      =		0
				while head:
								if head.val in d:
												d[head.val].append(__a		)
								else:
												a__				      =		[pos]
								a__				      =		head.next
								pos += 1
				a__				      =		pos - 1
				a__				      =		0
				for v in d.values():
								if len(__a		) % 2 != 0:
												middle += 1
								else:
												a__				      =		0
												for i in range(0 ,		len(__a		)		):
																if v[i] + v[len(__a		) - 1 - step] != checksum:
																				return False
																step += 1
								if middle > 1:
												return False
				return True

