C. Cat, Fox and Double Maximum

Fox loves permutations! She came up with the following problem and asked Cat to solve it:

You are given an even positive integer 𝑛 and a permutation† 𝑝 of length 𝑛.

The score of another permutation 𝑞 of length 𝑛 is the number of local maximums in the array 𝑎 of length 𝑛, where 𝑎𝑖=𝑝𝑖+𝑞𝑖 for all 𝑖 (1≤𝑖≤𝑛). In other words, the score of 𝑞 is the number of 𝑖 such that 1<𝑖<𝑛 (note the strict inequalities), 𝑎𝑖−1<𝑎𝑖, and 𝑎𝑖>𝑎𝑖+1 (once again, note the strict inequalities).

Find the permutation 𝑞 that achieves the maximum score for given 𝑛 and 𝑝. If there exist multiple such permutations, you can pick any of them.

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).