
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 vectors $p = (a_n, 2^n)$ and $q = (2^{n+1}, -a_{n+1})$, where $n \in \mathbb{N}^*$, and vectors $p$ and $q$ are perpendicular, with $a_1 = 1$.

(1) Find the general formula for the sequence $\{a_n\}$;  
(2) If the sequence $\{b_n\}$ satisfies $b_n = \log_2{a_n} + 1$, find the sum of the first $n$ terms $S_n$ of the sequence $\{a_n \cdot b_n\}$.

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