
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
In the geometric sequence $\{a_n\}$ where each term is positive, $a_1=2$, and $2a_1, a_3, 3a_2$ form an arithmetic sequence.

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

$(2)$ If the sequence $\{b_n\}$ satisfies $b_n=11-2\log_2{a_n}$, find the maximum value of the sum of the first $n$ terms of $\{b_n\}$, denoted as $T_n$.

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