
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
The sequence $\{b_n\}$ ($n \in \mathbb{N}^*$) is an increasing geometric sequence with $b_1 + b_3 = 17$ and $b_1b_3 = 16$. Additionally, let $a_n = \log_4 b_n + 2$.  
(1) Find the general formulas for the sequences $\{a_n\}$ and $\{b_n\}$.  
(2) If $a_1^2 + a_2 + a_3 + \dots + a_m \leq a_{66}$, find the maximum value of $m$.

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