OFM 2023 J1

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