
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
Given an arithmetic sequence $\{a_n\}$ with first term $a_1 = 1$ and common ratio $q > 0$, let $S_n$ denote the sum of its first $n$ terms. The sequence $\{S_1 + a_1, S_3 + a_3, S_2 + a_2\}$ forms an arithmetic progression.

(1) Find the general term formula for the sequence $\{a_n\}$.

(2) If the sequence $\{a_n\}$ satisfies $a_{n + 1} = \left(\frac{1}{2}\right)^{a_n b_n}$, let $T_n$ denote the sum of the first $n$ terms of the sequence $\{b_n\}$. If $T_n \geq m$ always holds, find the maximum value of $m$.

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