D. Permutation Game

Bodya and Sasha found a permutation 𝑝1,…,𝑝𝑛 and an array 𝑎1,…,𝑎𝑛. They decided to play a well-known "Permutation game".

A permutation of length 𝑛 is an array consisting of 𝑛 distinct integers from 1 to 𝑛 in arbitrary order. For example, [2,3,1,5,4] is a permutation, but [1,2,2] is not a permutation (2 appears twice in the array), and [1,3,4] is also not a permutation (𝑛=3 but there is 4 in the array).

Both of them chose a starting position in the permutation.

The game lasts 𝑘 turns. The players make moves simultaneously. On each turn, two things happen to each player:

• If the current position of the player is 𝑥 his score increases by 𝑎
• .Then the player either stays at his current position 𝑥 or moves from 𝑥 to 𝑝𝑥.

The winner of the game is the player with the higher score after exactly 𝑘 turns.

Knowing Bodya's starting position 𝑃𝐵 and Sasha's starting position 𝑃𝑆, determine who wins the game if both players are trying to win.