"""simple docstring"""






import sys




def 						SCREAMING_SNAKE_CASE					(  __UpperCAmelCase       )							->   Any:
				SCREAMING_SNAKE_CASE__		  =				len(__UpperCAmelCase       )
				SCREAMING_SNAKE_CASE__		  =				[[0 for x in range(__UpperCAmelCase       )] for x in range(__UpperCAmelCase       )]
				SCREAMING_SNAKE_CASE__		  =				[[0 for x in range(__UpperCAmelCase       )] for x in range(__UpperCAmelCase       )]

				for chain_length in range(2  ,					__UpperCAmelCase       ):
								for a in range(1  ,					n - chain_length + 1       ):
												SCREAMING_SNAKE_CASE__		  =				a + chain_length - 1

												SCREAMING_SNAKE_CASE__		  =				sys.maxsize
												for c in range(__UpperCAmelCase  ,					__UpperCAmelCase       ):
																SCREAMING_SNAKE_CASE__		  =				(
																    matrix[a][c] + matrix[c + 1][b] + array[a - 1] * array[c] * array[b]
																)
																if cost < matrix[a][b]:
																				SCREAMING_SNAKE_CASE__		  =				cost
																				SCREAMING_SNAKE_CASE__		  =				c
				return matrix, sol




def 						SCREAMING_SNAKE_CASE					(  __UpperCAmelCase  ,					__UpperCAmelCase  ,					__UpperCAmelCase       )							->   Union[str, Any]:
				if i == j:
								print("A" + str(__UpperCAmelCase       )  ,					end=" "       )
				else:
								print("("  ,					end=" "       )
								print_optiomal_solution(__UpperCAmelCase  ,					__UpperCAmelCase  ,					optimal_solution[i][j]       )
								print_optiomal_solution(__UpperCAmelCase  ,					optimal_solution[i][j] + 1  ,					__UpperCAmelCase       )
								print(")"  ,					end=" "       )




def 						SCREAMING_SNAKE_CASE					(  )							->   Any:
				SCREAMING_SNAKE_CASE__		  =				[30, 35, 15, 5, 10, 20, 25]
				SCREAMING_SNAKE_CASE__		  =				len(__UpperCAmelCase       )
				# Size of matrix created from above array will be
				# 30*35 35*15 15*5 5*10 10*20 20*25
				SCREAMING_SNAKE_CASE__  ,       SCREAMING_SNAKE_CASE__		  =				matrix_chain_order(__UpperCAmelCase       )

				print("No. of Operation required: " + str(matrix[1][n - 1]       )       )
				print_optiomal_solution(__UpperCAmelCase  ,					1  ,					n - 1       )


if __name__ == "__main__":
						main()





"""simple docstring"""






def 						SCREAMING_SNAKE_CASE					(  __UpperCAmelCase       )							->   List[str]:
				if not head:
								return True
				# split the list to two parts
				SCREAMING_SNAKE_CASE__  ,       SCREAMING_SNAKE_CASE__		  =				head.next, head
				while fast and fast.next:
								SCREAMING_SNAKE_CASE__		  =				fast.next.next
								SCREAMING_SNAKE_CASE__		  =				slow.next
				SCREAMING_SNAKE_CASE__		  =				slow.next
				SCREAMING_SNAKE_CASE__		  =				None  # Don't forget here! But forget still works!
				# reverse the second part
				SCREAMING_SNAKE_CASE__		  =				None
				while second:
								SCREAMING_SNAKE_CASE__		  =				second.next
								SCREAMING_SNAKE_CASE__		  =				node
								SCREAMING_SNAKE_CASE__		  =				second
								SCREAMING_SNAKE_CASE__		  =				nxt
				# compare two parts
				# second part has the same or one less node
				while node:
								if node.val != head.val:
												return False
								SCREAMING_SNAKE_CASE__		  =				node.next
								SCREAMING_SNAKE_CASE__		  =				head.next
				return True




def 						SCREAMING_SNAKE_CASE					(  __UpperCAmelCase       )							->   str:
				if not head or not head.next:
								return True

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

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

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

				return True




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




