There are $n$ houses numbered from $0$ to $n - 1$ arranged in a circle in clockwise order.

You start at house $0$ at time $0$. You are given two $0$ indexed integer arrays forward and backward, each of length $n$, and an integer array queries.

For each house $i$ (where $0 \leq i < n$):

• Traveling clockwise from house $i$ to house $(i + 1) \bmod n$ takes forward[i] seconds.

• Traveling counterclockwise from house $i$ to house $(i - 1 + n) \bmod n$ takes backward[i] seconds.

You need to visit the houses in the sequence given by queries. For each consecutive destination, you can choose to travel either clockwise or counterclockwise along the circle to minimize the travel time.

Return the minimum total time required to visit all houses in the specified order.

Constraints:

• $2 \leq n \leq 10^5$

• $n ==$ forward.length $==$ backward.length

• $1 \leq$ forward[i], backward[i] $\leq 10^5$

• $1 \leq$ queries.length $\leq 10^5$

• $0 \leq$ queries[i] $< n$

• queries[i] $\neq$ queries[i + 1]

• queries[0] $\neq 0$

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.