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 $\leq 10^5$

Now, let’s take a moment to make sure you’ve correctly understood the problem. The quiz below helps you check if you’re solving the correct problem:

We have a game for you to play. Rearrange the logical building blocks to develop a clearer understanding of how to solve this problem.

Implement your solution in the following coding playground.