
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 a non-zero common difference, where the first term $a_1=1$, and the terms $a_1, a_3, a_9$ form a geometric sequence.
$(1)$ Find the general formula for the sequence ${\{a_n\}}$.
$(2)$ Let another sequence ${\{b_n\}}$ be defined as $b_n = a_n + 2^{a_n}$. Compute the sum of the first $n$ terms of the sequence ${\{b_n\}}$, denoted by $T_n$.

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