
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
Let the sequence \(\{a_n\}\) be defined as follows:
\[ 
\begin{aligned}
& a_1 = 1, \quad a_2 = 3, \\
& a_n = 3a_{n-1} - a_{n-2} \quad (n \in \mathbb{Z}_+, n \geq 3)
\end{aligned}
\]

Is there a positive integer \(n\) such that \(2^{2016} \| a_n\), i.e., \(2^{2016} \mid a_n\) and \(2^{2017} \nmid a_n\)? If such an \(n\) exists, find the smallest positive integer \(n\); if not, provide a justification.

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