You are given an integer array differences of length n, and two integers lower and upper. A hidden integer array hidden of length n + 1 is considered valid if it satisfies both conditions:

The value at each index stays within the inclusive range from lower to upper.

For every index i in the range from $0$ to $n - 1$, the adjacent difference matches differences, meaning hidden[i + 1] - hidden[i] == differences[i].

Return the number of distinct valid arrays hidden that can be formed.

Constraints:

• n == differences.length

• $1 \leq$ n $\leq 10^5$

• $-10^5 \leq$ differences[i] $\leq 10^5$

• $-10^5 \leq$ lower $\leq$ upper $\leq 10^5$

Input: differences = [1, -3, 4]-List[int], lower = 1-int, upper = 6-int
Output: 2-int
Explanation: Let hidden[0] = x. The prefix sums of differences are [0, 1, -2, 2], so hidden = [x, x+1, x-2, x+2]. Keeping every value within [1, 6] requires x in [1, 6], [0, 5], [3, 8], and [-1, 4], whose intersection is x ∈ [3, 4], giving 2 valid arrays.

Input: differences = [3, 3, 3]-List[int], lower = 0-int, upper = 6-int
Output: 0-int
Explanation: The prefix sums are [0, 3, 6, 9], so hidden = [x, x+3, x+6, x+9], which spans 9 units from its smallest to its largest value. The allowed range [0, 6] only spans 6 units (x >= 0 and x + 9 <= 6 cannot both hold), so no valid hidden sequence exists.

Input: differences = [-2, -1, 2]-List[int], lower = -3-int, upper = 2-int
Output: 3-int
Explanation: The prefix sums are [0, -2, -3, -1], so hidden = [x, x-2, x-3, x-1]. The range constraints require x in [-3, 2], [-1, 4], [0, 5], and [-2, 3], whose intersection is x ∈ [0, 2], so the three starting values 0, 1, and 2 all work.

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