
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 sequence \(\{a_n\}\) with the general term
$$
a_n = \frac{1}{\sqrt{5}} \left( \left( \frac{1+\sqrt{5}}{2} \right)^n - \left( \frac{1-\sqrt{5}}{2} \right)^n \right) \quad (n \in \mathbb{Z}_{+}),
$$

let \(S_n = \binom{n}{1}a_1 + \binom{n}{2}a_2 + \cdots + \binom{n}{n}a_n\). Determine all positive integers \(n\) such that \(7\) divides \(S_n\).

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