<p>You are given a <strong>0-indexed</strong> array <code>nums</code> and an integer <code>target</code>.</p>

<p>A <strong>0-indexed</strong> array <code>infinite_nums</code> is generated by infinitely appending the elements of <code>nums</code> to itself.</p>

<p>Return <em>the length of the <strong>shortest</strong> subarray of the array </em><code>infinite_nums</code><em> with a sum equal to </em><code>target</code><em>.</em> If there is no such subarray return <code>-1</code>.</p>

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

<pre>
<strong>Input:</strong> nums = [1,2,3], target = 5
<strong>Output:</strong> 2
<strong>Explanation:</strong> In this example infinite_nums = [1,2,3,1,2,3,1,2,...].
The subarray in the range [1,2], has the sum equal to target = 5 and length = 2.
It can be proven that 2 is the shortest length of a subarray with sum equal to target = 5.
</pre>

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

<pre>
<strong>Input:</strong> nums = [1,1,1,2,3], target = 4
<strong>Output:</strong> 2
<strong>Explanation:</strong> In this example infinite_nums = [1,1,1,2,3,1,1,1,2,3,1,1,...].
The subarray in the range [4,5], has the sum equal to target = 4 and length = 2.
It can be proven that 2 is the shortest length of a subarray with sum equal to target = 4.
</pre>

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

<pre>
<strong>Input:</strong> nums = [2,4,6,8], target = 3
<strong>Output:</strong> -1
<strong>Explanation:</strong> In this example infinite_nums = [2,4,6,8,2,4,6,8,...].
It can be proven that there is no subarray with sum equal to target = 3.
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= target &lt;= 10<sup>9</sup></code></li>
</ul>

from typing import List

class Solution:
    def minSizeSubarray(self, nums: List[int], target: int) -> int:
        total_sum = sum(nums)
        n = len(nums)
        
        # If total_sum is larger than target, only need one segment of nums (just like normal array problem but consider prefix_sum % total_sum)
        if total_sum == 0:
            return -1
        min_length = float('inf')
        
        if target >= total_sum:
            full_rounds = target // total_sum
            target = target % total_sum
        else:
            full_rounds = 0
        
        # If target now is zero after reduction, we need full_rounds * size of 'nums'
        if target == 0:
            return full_rounds * n
        
        prefix_sum = 0
        prefix_map = {0: -1}  # To handle sum found exactly at index `i`
        
        # Consider up to two concatenated rounds of nums
        for i in range(2 * n):
            prefix_sum += nums[i % n]
            modded_target = (prefix_sum - target) % total_sum
            
            if modded_target in prefix_map:
                start_index = prefix_map[modded_target]
                length = i - start_index
                if length <= n:  # Length must not exceed a single round of nums
                    min_length = min(min_length, length)
            
            prefix_map[prefix_sum % total_sum] = i
        
        if min_length == float('inf'):
            return -1
        else:
            return min_length + full_rounds * n
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