<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)
        max_sum = 0
        
        for peak_index in range(n):
            adjusted_heights = [0] * n
            adjusted_heights[peak_index] = heights[peak_index]
            left_index = peak_index - 1
            right_index = peak_index + 1
            
            # Construct the left side of the mountain
            while left_index >= 0:
                if heights[left_index] <= adjusted_heights[left_index + 1]:
                    adjusted_heights[left_index] = heights[left_index]
                else:
                    adjusted_heights[left_index] = adjusted_heights[left_index + 1]
                left_index -= 1
            
            # Construct the right side of the mountain
            while right_index < n:
                if heights[right_index] <= adjusted_heights[right_index - 1]:
                    adjusted_heights[right_index] = heights[right_index]
                else:
                    adjusted_heights[right_index] = adjusted_heights[right_index - 1]
                right_index += 1
            
            current_sum = sum(adjusted_heights)
            max_sum = max(max_sum, current_sum)
        
        return max_sum
924