<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:
    kMod = 1_000_000_007
    n = len(nums)
    ans = 0
    prefix = list(itertools.accumulate(nums))

    def firstGreaterEqual(i: int) -> int:
      """Finds the first index j s.t.
         Mid = prefix[j] - prefix[i] >= left = prefix[i]
      """
      l = i + 1
      r = n - 1
      while l < r:
        m = (l + r) // 2
        if prefix[m] - prefix[i] >= prefix[i]:
          r = m
        else:
          l = m + 1
      return l

    def firstGreater(i: int) -> int:
      """Finds the first index k s.t.
         mid = prefix[k] - prefix[i] > right = prefix[-1] - prefix[k]
      """
      l = i + 1
      r = n - 1
      while l < r:
        m = (l + r) // 2
        if prefix[m] - prefix[i] > prefix[-1] - prefix[m]:
          r = m
        else:
          l = m + 1
      return l

    for i in range(n - 2):
      j = firstGreaterEqual(i)
      if j == n - 1:
        break
      mid = prefix[j] - prefix[i]
      right = prefix[-1] - prefix[j]
      if mid > right:
        continue
      k = firstGreater(i)
      ans = (ans + k - j) % kMod

    return ans
1403