Let $u_0, u_1, u_2, \ldots$ be integers such that $u_0 = 100$; $u_{k+2} \geq 2 + u_k$ for all $k \geq 0$; and $u_{\ell+5} \leq 5 + u_\ell$ for all $\ell \geq 0$. Find all possible values for the integer $u_{2023}$.

Let's think step by step and output the final answer within \boxed{}.
user