
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 general term of the sequence \(\{f_n\}\) is given by
\[
f_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right], \quad n \in \mathbb{Z}^{+}.
\]

Let \(S_n = \binom{n}{1} f_1 + \binom{n}{2} f_2 + \cdots + \binom{n}{n} f_n\). Find all positive integers \(n\) such that \(S_n\) is divisible by 8.

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