Ancient Egyptians are known to have understood difficult concepts in mathematics. The ancient Egyptian mathematician Ahmes liked to write a kind of arithmetic expressions on papyrus paper which he called as Ahmes arithmetic expression.

An Ahmes arithmetic expression can be defined as:   "d" is an Ahmes arithmetic expression, where d is a one-digit positive integer;  "(E_1 op E_2)" is an Ahmes arithmetic expression, where E_1 and E_2 are valid Ahmes arithmetic expressions (without spaces) and op is either plus ( + ) or minus ( - ).  For example 5, (1-1) and ((1+(2-3))-5) are valid Ahmes arithmetic expressions.

On his trip to Egypt, Fafa found a piece of papyrus paper having one of these Ahmes arithmetic expressions written on it. Being very ancient, the papyrus piece was very worn out. As a result, all the operators were erased, keeping only the numbers and the brackets. Since Fafa loves mathematics, he decided to challenge himself with the following task:

Given the number of plus and minus operators in the original expression, find out the maximum possible value for the expression on the papyrus paper after putting the plus and minus operators in the place of the original erased operators.


-----Input-----

The first line contains a string E (1 ≤ |E| ≤ 10^4) — a valid Ahmes arithmetic expression. All operators are erased and replaced with '?'.

The second line contains two space-separated integers P and M (0 ≤ min(P, M) ≤ 100) — the number of plus and minus operators, respectively. 

It is guaranteed that P + M =  the number of erased operators.


-----Output-----

Print one line containing the answer to the problem.


-----Examples-----
Input
(1?1)
1 0

Output
2

Input
(2?(1?2))
1 1

Output
1

Input
((1?(5?7))?((6?2)?7))
3 2

Output
18

Input
((1?(5?7))?((6?2)?7))
2 3

Output
16



-----Note-----  The first sample will be (1 + 1)  =  2.  The second sample will be (2 + (1 - 2))  =  1.  The third sample will be ((1 - (5 - 7)) + ((6 + 2) + 7))  =  18.  The fourth sample will be ((1 + (5 + 7)) - ((6 - 2) - 7))  =  16.

Difficulty: HARD
Tags: Tree algorithms

Dynamic programming

Graph traversal
Skill types: Dynamic programming

Starter code:

Solutions:
from math import inf

class Node:

	def __init__(self, parent=None, leftExp=None, rightExp=None, signQ=0):
		(self.parent, self.leftExp, self.rightExp, self.signQ) = (parent, leftExp, rightExp, signQ)

	def __str__(self):
		return 'Node'
memo = {}

def Memoize(node, p, maxValue, minValue):
	if not node in memo:
		memo.update({node: {p: [maxValue, minValue]}})
	else:
		memo[node].update({p: [maxValue, minValue]})

def ExpMaxValue(root: Node, p):
	m = root.signQ - p
	if root.signQ == 0:
		return [root.value, root.value]
	if root in memo:
		if p in memo[root]:
			return memo[root][p]
	if m == 0:
		value = ExpMaxValue(root.leftExp, root.leftExp.signQ)[0] + ExpMaxValue(root.rightExp, root.rightExp.signQ)[0]
		Memoize(root, p, value, value)
		return [value, value]
	if p == 0:
		value = ExpMaxValue(root.leftExp, 0)[0] - ExpMaxValue(root.rightExp, 0)[0]
		Memoize(root, p, value, value)
		return [value, value]
	maxValue = -inf
	minValue = inf
	if m >= p:
		for pQMid in range(2):
			pQLeftMin = min(p - pQMid, root.leftExp.signQ)
			for pQLeft in range(pQLeftMin + 1):
				if root.leftExp.signQ - pQLeft + (1 - pQMid) > m:
					continue
				resLeft = ExpMaxValue(root.leftExp, pQLeft)
				resRight = ExpMaxValue(root.rightExp, p - pQMid - pQLeft)
				if pQMid == 1:
					maxValue = max(resLeft[0] + resRight[0], maxValue)
					minValue = min(resLeft[1] + resRight[1], minValue)
				else:
					maxValue = max(resLeft[0] - resRight[1], maxValue)
					minValue = min(resLeft[1] - resRight[0], minValue)
	else:
		for mQMid in range(2):
			mQLeftMin = min(m - mQMid, root.leftExp.signQ)
			for mQLeft in range(mQLeftMin + 1):
				pQLeft = root.leftExp.signQ - mQLeft
				if pQLeft + (1 - mQMid) > p:
					continue
				resLeft = ExpMaxValue(root.leftExp, pQLeft)
				resRight = ExpMaxValue(root.rightExp, p - (1 - mQMid) - pQLeft)
				if mQMid == 0:
					maxValue = max(resLeft[0] + resRight[0], maxValue)
					minValue = min(resLeft[1] + resRight[1], minValue)
				else:
					maxValue = max(resLeft[0] - resRight[1], maxValue)
					minValue = min(resLeft[1] - resRight[0], minValue)
	Memoize(root, p, int(maxValue), int(minValue))
	return [int(maxValue), int(minValue)]

def PrintNodes(root: Node):
	if root.signQ != 0:
		leftExp = root.leftExp if root.leftExp.signQ != 0 else root.leftExp.value
		rightExp = root.rightExp if root.rightExp.signQ != 0 else root.rightExp.value
		check = root.signQ == root.leftExp.signQ + root.rightExp.signQ + 1
		print(root.signQ, root.parent, leftExp, rightExp, check)
		PrintNodes(root.leftExp)
		PrintNodes(root.rightExp)

def main():
	exp = str(input())
	if len(exp) == 1:
		print(exp)
		return
	cNode = Node()
	isRight = False
	for i in range(1, len(exp) - 1):
		if exp[i] == '(':
			if not isRight:
				cNode.leftExp = Node(cNode)
				cNode = cNode.leftExp
			else:
				cNode.rightExp = Node(cNode)
				cNode = cNode.rightExp
			isRight = False
		elif exp[i] == '?':
			isRight = True
			cNode.signQ += 1
		elif exp[i] == ')':
			if cNode.parent != None:
				cNode.parent.signQ += cNode.signQ
			cNode = cNode.parent
			isRight = False
		elif not isRight:
			cNode.leftExp = Node(cNode)
			cNode.leftExp.value = int(exp[i])
		else:
			cNode.rightExp = Node(cNode)
			cNode.rightExp.value = int(exp[i])
	ss = str(input()).split()
	(p, m) = (int(ss[0]), int(ss[1]))
	print(ExpMaxValue(cNode, p)[0])
main()