
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 $\left\{a_n\right\}$ with the sum of the first $n$ terms denoted as $S_n$, and $a_3 = 5$, $S_3 = 9$,

$(1)$ Find the general formula for the sequence $\left\{a_n\right\}$;

$(2)$ Let $\left\{b_n\right\}$ be a geometric sequence with the sum of the first $n$ terms denoted as $T_n$, if $q > 0$ and $b_3 = a_5, T_3 = 13$, find $T_n$.

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