
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]


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Given a sequence $\{a_n\}$ where all terms are positive, the sum of the first $n$ terms is denoted as $S_n$, and it satisfies the equation $4S_n = (a_n + 1)^2$.

(1) Find the general term formula for $\{a_n\}$;
(2) Let $b_n = 2^{a_n}$, and denote the sum of the first $n$ terms of the sequence $\{a_n \cdot b_n\}$ as $T_n$. Find $T_n$.

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