Alice and Bob are playing a turn based game on a circular field surrounded by flowers. There are $x$ flowers clockwise and $y$ flowers counterclockwise around the circle.

The rules of the game are as follows:

• Alice moves first.

• In each turn, the current player must pick $1$ flower from either the clockwise direction or the counterclockwise direction.

• The player who picks the last flower wins the game.

Given two integers n and m, determine the number of pairs $(x, y)$ such that $1 \le x \le$ n, $1 \le y \le$ m, and Alice wins the game assuming both players play optimally.

Constraints:

• $1 \le$ n, m $\le 10^5$

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.