<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
        # prime factorization for each elem
        # dict to record first and last pos for each prime: [left, right)
        mp = defaultdict(list)
        for i, v in enumerate(nums):
            d = 2
            while d * d <= v:
                if v % d == 0:
                    # d is a prime factor
                    if len(mp[d]) == 0:
                        mp[d].append(i)
                    elif len(mp[d]) == 1:
                        mp[d].append(i)
                    else:
                        mp[d][1] = i
                    v //= d
                    while v % d == 0:
                        v //= d
                d += 1
            if v > 1: # if remaining > 1, then it is a prime, e.g., 14 = 2 * 7, now v = 7
                if len(mp[v]) == 0:
                    mp[v].append(i)
                elif len(mp[v]) == 1:
                    mp[v].append(i)
                else:
                    mp[v][1] = i

        intervals = []
        for k, v in mp.items():
            if len(v) == 2:
                intervals.append(v)
        # first and last index that can divide: cut at first not fine, cut at last is fine
        intervals = sorted(intervals)

        if not intervals or intervals[0][0] > 0:
            return 0
        
        mx = intervals[0][1]
        for i in range(1, len(intervals)):
            if intervals[i][0] > mx:
                return mx
            mx = max(mx, intervals[i][1])
        
        if mx < len(nums) - 1:
            return mx
        return -1
19100