<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:
        def eratosthenes_sieve(n):
            yield 2
            is_prime = [True] * (n + 1)
            for prime in range(3, n + 1, 2):
                if is_prime[prime]:
                    yield prime
                    for elem in range(prime ** 2, n + 1, 2 * prime):
                        is_prime[elem] = False            
        
        primes = list(eratosthenes_sieve(max(nums)))
        primes_set = set(primes)

        @cache
        def get_unique_primes(elem):
            result = set()
            for prime in primes:
                while elem % prime == 0:
                    result.add(prime)
                    elem //= prime
                if elem == 1:
                    break
                if elem in primes_set:  # heuristics 1
                    result.add(elem)
                    break
            return result
        
        unique_primes = [
            get_unique_primes(elem)
            for elem in nums
        ]

        prefix = set()
        suffix = Counter()
        for up in unique_primes:
            suffix.update(up)
        
        for i in range(len(nums) - 1):
            exist_common_elem = False  # heuristics 2
            for prime in unique_primes[i]:
                if suffix[prime] == 1:
                    del suffix[prime]
                    prefix.discard(prime)
                else:
                    suffix[prime] -= 1
                    prefix.add(prime)
                    exist_common_elem = True
            
            if exist_common_elem:
                continue
            elif prefix.isdisjoint(suffix):
                return i

        return -1

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