<p>You are given a <strong>0-indexed</strong> integer array <code>nums</code>. For each index <code>i</code> (<code>1 &lt;= i &lt;= nums.length - 2</code>) the <strong>beauty</strong> of <code>nums[i]</code> equals:</p>

<ul>
	<li><code>2</code>, if <code>nums[j] &lt; nums[i] &lt; nums[k]</code>, for <strong>all</strong> <code>0 &lt;= j &lt; i</code> and for <strong>all</strong> <code>i &lt; k &lt;= nums.length - 1</code>.</li>
	<li><code>1</code>, if <code>nums[i - 1] &lt; nums[i] &lt; nums[i + 1]</code>, and the previous condition is not satisfied.</li>
	<li><code>0</code>, if none of the previous conditions holds.</li>
</ul>

<p>Return<em> the <strong>sum of beauty</strong> of all </em><code>nums[i]</code><em> where </em><code>1 &lt;= i &lt;= nums.length - 2</code>.</p>

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

<pre>
<strong>Input:</strong> nums = [1,2,3]
<strong>Output:</strong> 2
<strong>Explanation:</strong> For each index i in the range 1 &lt;= i &lt;= 1:
- The beauty of nums[1] equals 2.
</pre>

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

<pre>
<strong>Input:</strong> nums = [2,4,6,4]
<strong>Output:</strong> 1
<strong>Explanation:</strong> For each index i in the range 1 &lt;= i &lt;= 2:
- The beauty of nums[1] equals 1.
- The beauty of nums[2] equals 0.
</pre>

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

<pre>
<strong>Input:</strong> nums = [3,2,1]
<strong>Output:</strong> 0
<strong>Explanation:</strong> For each index i in the range 1 &lt;= i &lt;= 1:
- The beauty of nums[1] equals 0.
</pre>

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

<ul>
	<li><code>3 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>5</sup></code></li>
</ul>

class Solution:
    def sumOfBeauties(self, nums: List[int]) -> int:
        b, a, res = float("-inf"), float("inf"), 0
        prefix, suffix = [], []

        for i in range(0, len(nums)):
            prefix.append(b)
            if nums[i] > b: b = nums[i]
        
        for i in range(len(nums)-1, -1, -1):
            suffix.append(a)
            if nums[i] < a: a = nums[i]
        suffix = suffix[::-1]

        for i in range(1, len(nums)-1):
            if prefix[i] < nums[i] and suffix[i] > nums[i]: res += 2
            elif nums[i-1] < nums[i] and nums[i+1] > nums[i]: res += 1

        return res


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