
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 a non-zero common difference arithmetic sequence \( \{ a_n \} \) satisfying \( a_1 = 5 \), and \( a_3 \), \( a_6 \), \( a_{11} \) form a geometric sequence.

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

(2) Let \( b_n = a_n \cdot 3^{n-1} \), find the sum of the first \( n \) terms of the sequence \( \{ b_n \} \), denoted as \( S_n \).

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