
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 the sequence $\{a_n\}$ where $a_1=2, a_2=1, a_{n+2}= \begin{cases} \frac{2a_{n+1}}{a_n}, & a_{n+1}\geqslant 2 \\ \frac{4}{a_n}, & a_{n+1} < 2 \end{cases} (n\in\mathbb{N}^*), S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, find $S_{2016}$.

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