
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 $\{a_n\}$ satisfies $\frac{a_n + a_{n-1} + 2}{3} = \frac{(-1)^n + 1}{3}a_{n-1} + n$ for $n \geq 2$, calculate the sum of the first 16 terms of the sequence $\{a_n\}$.

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