<p>A split of an integer array is <strong>good</strong> if:</p>

<ul>
	<li>The array is split into three <strong>non-empty</strong> contiguous subarrays - named <code>left</code>, <code>mid</code>, <code>right</code> respectively from left to right.</li>
	<li>The sum of the elements in <code>left</code> is less than or equal to the sum of the elements in <code>mid</code>, and the sum of the elements in <code>mid</code> is less than or equal to the sum of the elements in <code>right</code>.</li>
</ul>

<p>Given <code>nums</code>, an array of <strong>non-negative</strong> integers, return <em>the number of <strong>good</strong> ways to split</em> <code>nums</code>. As the number may be too large, return it <strong>modulo</strong> <code>10<sup>9 </sup>+ 7</code>.</p>

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

<pre>
<strong>Input:</strong> nums = [1,1,1]
<strong>Output:</strong> 1
<strong>Explanation:</strong> The only good way to split nums is [1] [1] [1].</pre>

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

<pre>
<strong>Input:</strong> nums = [1,2,2,2,5,0]
<strong>Output:</strong> 3
<strong>Explanation:</strong> There are three good ways of splitting nums:
[1] [2] [2,2,5,0]
[1] [2,2] [2,5,0]
[1,2] [2,2] [5,0]
</pre>

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

<pre>
<strong>Input:</strong> nums = [3,2,1]
<strong>Output:</strong> 0
<strong>Explanation:</strong> There is no good way to split nums.</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>0 &lt;= nums[i] &lt;= 10<sup>4</sup></code></li>
</ul>

class Solution:
    def waysToSplit(self, nums: List[int]) -> int:
        def lower_bound(lo, hi, target):
            while lo < hi:
                mid = lo + (hi - lo) // 2
                if psa[mid] >= 2 * target:
                    hi = mid
                else:
                    lo = mid + 1
            if psa[lo] >= 2 * target:
                return lo
            else:
                return None

        def upper_bound(lo, hi, target):
            while lo < hi:
                mid = lo + (hi - lo + 1) // 2
                if psa[n - 1] - psa[mid] >= psa[mid] - target:
                    lo = mid
                else:
                    hi = mid - 1
            if psa[n - 1] - psa[lo] >= psa[lo] - target:
                return lo
            else:
                return None
        

        # iterate through left then binsearch using PSA 
        n = len(nums)
        psa = [nums[0]]
        for i in range(1, n):
            psa.append(nums[i] + psa[-1])
        
        ans = 0
        for i in range(n - 2):
            if 3*psa[i] > psa[n - 1]:
                break
            target = psa[i]
            lower_index = lower_bound(i + 1, n - 2, target)
            upper_index = upper_bound(i + 1, n - 2, target)
            if lower_index is not None and upper_index is not None and lower_index <= upper_index:
                ans += upper_index - lower_index + 1
        return ans % (10**9+7)
1222