<p>You are given a <strong>0-indexed</strong> array <code>maxHeights</code> of <code>n</code> integers.</p>

<p>You are tasked with building <code>n</code> towers in the coordinate line. The <code>i<sup>th</sup></code> tower is built at coordinate <code>i</code> and has a height of <code>heights[i]</code>.</p>

<p>A configuration of towers is <strong>beautiful</strong> if the following conditions hold:</p>

<ol>
	<li><code>1 &lt;= heights[i] &lt;= maxHeights[i]</code></li>
	<li><code>heights</code> is a <strong>mountain</strong> array.</li>
</ol>

<p>Array <code>heights</code> is a <strong>mountain</strong> if there exists an index <code>i</code> such that:</p>

<ul>
	<li>For all <code>0 &lt; j &lt;= i</code>, <code>heights[j - 1] &lt;= heights[j]</code></li>
	<li>For all <code>i &lt;= k &lt; n - 1</code>, <code>heights[k + 1] &lt;= heights[k]</code></li>
</ul>

<p>Return <em>the <strong>maximum possible sum of heights</strong> of a beautiful configuration of towers</em>.</p>

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

<pre>
<strong>Input:</strong> maxHeights = [5,3,4,1,1]
<strong>Output:</strong> 13
<strong>Explanation:</strong> One beautiful configuration with a maximum sum is heights = [5,3,3,1,1]. This configuration is beautiful since:
- 1 &lt;= heights[i] &lt;= maxHeights[i]  
- heights is a mountain of peak i = 0.
It can be shown that there exists no other beautiful configuration with a sum of heights greater than 13.</pre>

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

<pre>
<strong>Input:</strong> maxHeights = [6,5,3,9,2,7]
<strong>Output:</strong> 22
<strong>Explanation:</strong> One beautiful configuration with a maximum sum is heights = [3,3,3,9,2,2]. This configuration is beautiful since:
- 1 &lt;= heights[i] &lt;= maxHeights[i]
- heights is a mountain of peak i = 3.
It can be shown that there exists no other beautiful configuration with a sum of heights greater than 22.</pre>

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

<pre>
<strong>Input:</strong> maxHeights = [3,2,5,5,2,3]
<strong>Output:</strong> 18
<strong>Explanation:</strong> One beautiful configuration with a maximum sum is heights = [2,2,5,5,2,2]. This configuration is beautiful since:
- 1 &lt;= heights[i] &lt;= maxHeights[i]
- heights is a mountain of peak i = 2. 
Note that, for this configuration, i = 3 can also be considered a peak.
It can be shown that there exists no other beautiful configuration with a sum of heights greater than 18.
</pre>

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

<ul>
	<li><code>1 &lt;= n == maxHeights.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= maxHeights[i] &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution:
    def maximumSumOfHeights(self, maxHeights: List[int]) -> int:

        def staircase(heights, N):
            ret = []
            curr = 0
            hs = []
            ls = []
            for curr_height in heights:
                le = 1
                while hs and curr_height <= hs[-1]:
                    curr -= (hs[-1] - curr_height) * ls[-1]
                    le += ls[-1]
                    hs.pop()
                    ls.pop()
                hs.append(curr_height)
                ls.append(le)
                curr += curr_height
                ret.append(curr)
            return ret
        
        N = len(maxHeights)
        fwd = staircase(maxHeights, N)
        bck = staircase(reversed(maxHeights), N)[::-1]
        ret = max(fwd[N-1], bck[0])
        for i in range(N-1):
            ret = max(ret, fwd[i]+bck[i+1])
        return ret
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