A genetic algorithm improves a population of candidate solutions the way evolution does: select fit parents, combine them, and add random mutations. Here the candidates are bit strings and fitness is the number of 1s (the OneMax problem).

To make it testable, every random choice is given. With N strings (N is even), build N children in pairs. For pair k = 0, 1, …, N/2 - 1:

1. Selection: parent1 wins tournament tournaments[2k] and parent2 wins tournaments[2k + 1]. A tournament [i, j] is won by whichever of population[i] and population[j] has more 1s; on a tie, population[i] wins.
2. Crossover at c = crossover_points[k]: child 2k = parent1[0:c] + parent2[c:] and child 2k + 1 = parent2[0:c] + parent1[c:].

Then apply each mutation [child, position] in order by flipping that bit of that child.

Implement ga_generation(population, tournaments, crossover_points, mutations).

Input

population: N bit strings of equal length L.

tournaments: N pairs of population indices.

crossover_points: N/2 cut points, each between 1 and L - 1.

mutations: [child, position] pairs (may be empty).

Output

Return the N children in order.

Input:
mutations = [
    [0, 0]
]
population = ["0000", "1111", "1100", "0011"]
tournaments = [
    [0, 1],
    [2, 3],
    [1, 2],
    [0, 3]
]
crossover_points = [2, 1]

Expected output:
["0100", "1111", "1011", "0111"]

Input:
mutations = []
population = ["10", "01"]
tournaments = [
    [0, 1],
    [1, 0]
]
crossover_points = [1]

Expected output:
["11", "00"]