
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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  $u_{2023}$ .

Present the answer in LaTex format: \boxed{Your answer}
user