<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>

class Solution:
    def minSizeSubarray(self, nums: List[int], target: int) -> int:
        sm = sum(nums)
        pref = list(accumulate(nums))
        res = inf
        for i in range(len(nums)):
            if nums[i]>target:
                continue
            elif nums[i]==target:
                return 1
            rem = pref[-1] if i==0 else pref[-1]-pref[i-1]
            if rem>=target:
                lp = i
                rp = len(nums) - 1
                while lp<=rp:
                    mid = (lp+rp)>>1
                    curr = pref[mid]
                    if i>0:
                        curr-=pref[i-1]
                    if curr>=target:
                        rp = mid - 1
                    else:
                        lp = mid + 1
                curr = pref[lp]
                if i>0: 
                    curr-=pref[i-1]
                #print(i,curr)
                if curr==target:
                    res = min(res, lp - i + 1)
            else:
        
                curr = len(nums) - i
                need = (target - rem)//sm
                curr += len(nums)*need
                rem+=sm*need
                rest = target - rem
                if rest==0:
                    res=min(res,curr)
                if rem!=target:
                    lp = 0
                    rp = len(nums) - 1
                    while lp<=rp:
                        mid = (lp+rp)>>1
                        if pref[mid]>=rest:
                            rp = mid - 1
                        else:
                            lp = mid + 1
                    rem += pref[lp]
                    if rem==target:
                        res = min(res,curr+lp+1)
        return res if res!=inf else -1
                
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