
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 $\{a_n\}$, if for any $n \in \mathbb{N}^*$, it satisfies $a_n + a_{n+1} + a_{n+2} + a_{n+3} = s$ (where $s$ is a constant), then the sequence is called a 4th-order equal sum sequence, where $s$ is the 4th-order public sum. If it satisfies $a_n \cdot a_{n+1} \cdot a_{n+2} = t$ (where $t$ is a constant), then the sequence is called a 3rd-order equal product sequence, where $t$ is the 3rd-order public product. It is known that the sequence $\{p_n\}$ is a 4th-order equal sum sequence with the first term being 1, and it satisfies $\frac{p_4}{p_3} = \frac{p_3}{p_2} = \frac{p_2}{p_1} = 2$. The sequence $\{q_n\}$ is a 3rd-order equal product sequence with a public product of 1, and $q_1 = q_2 = -1$. If $S_n$ is the sum of the first $n$ terms of the sequence $\{p_n \cdot q_n\}$, then $S_{2016} =$ _____.

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