
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 that the sequence $\{a_n\}$ is a geometric sequence with the first term $a_1=1$ and common ratio $q > 0$, the sum of the first $n$ terms is $S_n$, and $S_1+a_1$, $S_3+a_3$, $S_2+a_2$ form an arithmetic sequence.

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

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

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