
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
The geometric sequence $\{a_n\}$ satisfies: $a_1+a_6=11$, $a_3 \cdot a_4 = \frac{32}{9}$, and the common ratio $q \in (0, 1)$.

(1) Find the general formula for the sequence $\{a_n\}$.
(2) If the sum of the first $n$ terms of this sequence, $S_n = 21$, find the value of $n$.

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