
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  $\{b_n\}_{n\geq 1}^{\infty}$  be a sequence of positive integers. The sequence  $\{a_n\}_{n\geq 1}^{\infty}$  is defined as follows:  $a_1$  is a fixed positive integer and 
\[a_{n+1}=a_n^{b_n}+1 ,\qquad \forall n\geq 1.\]
Find all positive integers  $m\geq 3$  with the following property: If the sequence  $\{a_n\mod m\}_{n\geq 1 }^{\infty}$  is eventually periodic, then there exist positive integers  $q,u,v$  with  $2\leq q\leq m-1$ , such that the sequence  $\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}$  is purely periodic.

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