
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 the sum of the first $n$ terms denoted as $S_n$, and $a_3=5$, $S_{15}=225$.

$(1)$ Find the general term formula $a_n$ of the sequence $\{a_n\}$;

$(2)$ Let $b_n = {2}^{an} + 2n$, find the sum $T_n$ of the first $n$ terms of the sequence $\{b_n\}$.

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