Make Permutation

Chef has an array $A$ of size $N$. He wants to make a permutation$^{\dagger}$ using this array. Find whether there exists an array $B$ consisting of $N$ non-negative integers, such that the array $C$ constructed as $C_i = A_i + B_i$ is a permutation. $^{\dagger}$ A permutation of size $N$ is an array of $N$ distinct elements in the range $[1, N]$. For example, $[4, 2, 1, 3]$ is a permutation of