<p>You are given a <strong>0-indexed</strong> integer array <code>nums</code> of length <code>n</code>.</p>

<p>A <strong>split</strong> at an index <code>i</code> where <code>0 &lt;= i &lt;= n - 2</code> is called <strong>valid</strong> if the product of the first <code>i + 1</code> elements and the product of the remaining elements are coprime.</p>

<ul>
	<li>For example, if <code>nums = [2, 3, 3]</code>, then a split at the index <code>i = 0</code> is valid because <code>2</code> and <code>9</code> are coprime, while a split at the index <code>i = 1</code> is not valid because <code>6</code> and <code>3</code> are not coprime. A split at the index <code>i = 2</code> is not valid because <code>i == n - 1</code>.</li>
</ul>

<p>Return <em>the smallest index </em><code>i</code><em> at which the array can be split validly or </em><code>-1</code><em> if there is no such split</em>.</p>

<p>Two values <code>val1</code> and <code>val2</code> are coprime if <code>gcd(val1, val2) == 1</code> where <code>gcd(val1, val2)</code> is the greatest common divisor of <code>val1</code> and <code>val2</code>.</p>

<p>&nbsp;</p>
<p><strong>Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/12/14/second.PNG" style="width: 450px; height: 211px;" />
<pre>
<strong>Input:</strong> nums = [4,7,8,15,3,5]
<strong>Output:</strong> 2
<strong>Explanation:</strong> The table above shows the values of the product of the first i + 1 elements, the remaining elements, and their gcd at each index i.
The only valid split is at index 2.
</pre>

<p><strong>Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/12/14/capture.PNG" style="width: 450px; height: 215px;" />
<pre>
<strong>Input:</strong> nums = [4,7,15,8,3,5]
<strong>Output:</strong> -1
<strong>Explanation:</strong> The table above shows the values of the product of the first i + 1 elements, the remaining elements, and their gcd at each index i.
There is no valid split.
</pre>

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

<ul>
	<li><code>n == nums.length</code></li>
	<li><code>1 &lt;= n &lt;= 10<sup>4</sup></code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>6</sup></code></li>
</ul>

class Solution:
    def findValidSplit(self, nums: List[int]) -> int:
        n = len(nums)
        if n == 1:
            return -1
        if nums[0] == 1:
            return 0
            
        prime_factors = [[] for _ in range(n)]
        for i, num in enumerate(nums):
            # find all prime factors of num
            j = 2
            temp = num
            while j * j <= temp:
                if temp % j == 0:
                    prime_factors[i].append(j)
                    while temp % j == 0:
                        temp //= j
                j += 1
            if temp > 1:
                prime_factors[i].append(temp)

        prime_to_min_idx = {}
        prime_to_max_idx = {}
        for i in range(n):
            for prime in prime_factors[i]:
                if prime not in prime_to_min_idx.keys():
                    prime_to_min_idx[prime] = i

                prime_to_max_idx[prime] = i
        
        right = list(set([v for v in prime_to_max_idx.values()]))
        right.sort()
        pq = []
        for prime, min_idx in prime_to_min_idx.items():
            heapq.heappush(pq, (min_idx, 1))
            heapq.heappush(pq, (prime_to_max_idx[prime], -1))
        
        temp = 0
        idx = 0
        ans = -1
        while idx < len(right) and right[idx] < n - 1:
            while len(pq) > 0 and pq[0][0] <= right[idx]:
                _, increment = heapq.heappop(pq)
                temp += increment
            
            if temp == 0:
                ans = right[idx]
                break
            
            idx += 1
        
        return ans

        
        

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