<p>You are given an array <code>heights</code> of <code>n</code> integers representing the number of bricks in <code>n</code> consecutive towers. Your task is to remove some bricks to form a <strong>mountain-shaped</strong> tower arrangement. In this arrangement, the tower heights are non-decreasing, reaching a maximum peak value with one or multiple consecutive towers and then non-increasing.</p>

<p>Return the <strong>maximum possible sum</strong> of heights of a mountain-shaped tower arrangement.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">heights = [5,3,4,1,1]</span></p>

<p><strong>Output:</strong> <span class="example-io">13</span></p>

<p><strong>Explanation:</strong></p>

<p>We remove some bricks to make <code>heights =&nbsp;[5,3,3,1,1]</code>, the peak is at index 0.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">heights = [6,5,3,9,2,7]</span></p>

<p><strong>Output:</strong> <span class="example-io">22</span></p>

<p><strong>Explanation:</strong></p>

<p>We remove some bricks to make <code>heights =&nbsp;[3,3,3,9,2,2]</code>, the peak is at index 3.</p>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">heights = [3,2,5,5,2,3]</span></p>

<p><strong>Output:</strong> <span class="example-io">18</span></p>

<p><strong>Explanation:</strong></p>

<p>We remove some bricks to make <code>heights = [2,2,5,5,2,2]</code>, the peak is at index 2 or 3.</p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n == heights.length &lt;= 10<sup>3</sup></code></li>
	<li><code>1 &lt;= heights[i] &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution:
    def maximumSumOfHeights(self, heights: List[int]) -> int:
        n = len(heights)
        left = [0]*n
        right = [0]*n

        stack = []
        for i in range(n):
            while stack and heights[stack[-1]] >= heights[i]:
                stack.pop()
            left[i] = i - (stack[-1] if stack else -1)
            stack.append(i)
        
        stack = []
        for i in range(n-1, -1, -1):
            while stack and heights[stack[-1]] >= heights[i]:
                stack.pop()
            right[i] = (stack[-1] if stack else n) - i
            stack.append(i)

        def getLeftMax(i):
            if i < 0: 
                return 0
            return (left[i] * heights[i]) + getLeftMax(i - left[i])
        
        ## For a given position, find its contribution towards its right such that it maintains the mountain property
        def getRightMax(i):
            if i >= n: 
                return 0
            return (right[i] * heights[i]) + getRightMax(i + right[i])
            
        ## Finally, consider every height as the peak of the mountain and calculate the sum
        maxSum = 0
        for i in range(n):
            curMax = heights[i]
            leftMax = getLeftMax(i)
            rightMax = getRightMax(i)
            maxSum = max(maxSum, leftMax + rightMax - curMax) ## curMax gets counted twice
        return maxSum

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