Problem Statement

Amazon Games is organizing a tournament of pair games where two teams of two players each compete against one another.

There are two groups, group1 and group2, each of size n. The skill levels of the ith players of the groups are group1[i] and group2[i], respectively. For each pair of indices (i, j) where 0 <= i < j < n, the pair of players (group1[i], group1[j]) competes in the pair game with (group2[i], group2[j]).

The winner of the game is:

• group1 if group1[i] + group1[j] > group2[i] + group2[j]
• group2 if group1[i] + group1[j] < group2[i] + group2[j]

If the sums are equal, there is no winner.

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Task

Write a function countGamesWonByGroup1(group1, group2) that computes the number of games won by group1. Since the answer can be large, return the result modulo 10^9 + 7.

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Function Signature

def countGamesWonByGroup1(group1: list[int], group2: list[int]) -> int:
    pass

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Input

• group1: A list of integers representing the skills of the players in group1.
• group2: A list of integers representing the skills of the players in group2.

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Output

• An integer representing the number of games won by group1, modulo 1000000007.

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Example

Example 1:
Input:

n = 3
group1 = [1, 2, 3]
group2 = [3, 2, 1]

Output:

1

Explanation:
The pairs and outcomes are:

• (0, 1) -> group1 = [1, 2], group2 = [3, 2], winner = group2
• (0, 2) -> group1 = [1, 3], group2 = [3, 1], no winner
• (1, 2) -> group1 = [2, 3], group2 = [2, 1], winner = group1

group1 wins one game, so the answer is 1.

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Example 2:
Input:

n = 3
group1 = [1, 3, 2]
group2 = [1, 3, 2]

Output:

0

Explanation:
All skills of both groups at respective indices are equal, so no one wins any games.

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Constraints

• 2 <= n <= 100000
• 1 <= group1[i], group2[i] <= 1000000000

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Notes

1. Iterate through all possible pairs (i, j) where 0 <= i < j < n to calculate the outcome for each game.
2. Use modulo 1000000007 to avoid integer overflow.
3. Optimize the solution to handle large values of n efficiently.

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