<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):
            # Initialize the sum with the peak value
            current_sum = heights[peak_index]
            
            # Calculate the non-decreasing sum from the start to the peak
            current_height = heights[peak_index]
            for i in range(peak_index - 1, -1, -1):
                current_height = min(current_height, heights[i])
                current_sum += current_height

            # Calculate the non-increasing sum from the peak to the end
            current_height = heights[peak_index]
            for i in range(peak_index + 1, n):
                current_height = min(current_height, heights[i])
                current_sum += current_height

            # Update the maximum sum
            max_sum = max(max_sum, current_sum)

        return max_sum

# Example usage:
solution = Solution()
print(solution.maximumSumOfHeights([5, 3, 4, 1, 1]))  # Output: 13
print(solution.maximumSumOfHeights([6, 5, 3, 9, 2, 7]))  # Output: 22
print(solution.maximumSumOfHeights([3, 2, 5, 5, 2, 3]))  # Output: 18
1376