After delivering all the Christmas presents, Santa finally have some leisure time to do Maths, which is his favourite hobby. Santa proposes two new sequences: Christmas sequence and Santa sequence. Numbers in the Christmas sequence are known as Christmas numbers. The Christmas sequence is defined as: $C_0 = 0, C_1 = 1, C_{n+1} = 2022C_n + C_{n-1}$ for $n \ge1$. The Santa sequence is defined as: $S_0 = 2, S_1 = 2022, S_{n+1} = 2022S_n + S_{n-1}$ for $n \ge1$. Santa finds $4043$ children and labels them from $1$ to $4043$. He asks the $n^{th}$ child to express $$C_1S_{2023} + C_2S_{2024} + ... + C_{2022}S_{4044}$$as a sum of $n$ non-zero Christmas numbers. Those who can do so can get an extra gift, which is a cute Christmas frog. Amongst the $4043$ children, who can potentially get an extra gift? Prove your claim.