
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 $\{a_n\}$ where each term is positive, the first term is $a_1=2$, and $4a_1$ is the arithmetic mean of $2a_2$ and $a_3$.

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

(2) If $b_n=a_n\cdot\log_2{a_n}$ and $S_n=b_1+b_2+\cdots+b_n$, find $S_n$.

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