
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 $y=f(x)$ be an even function defined on the set of real numbers, $R$, and it satisfies $f(x+4)=f(4-x)$ for all $x\in R$. Given that $f(x)=x$ for $x\in [0,4]$ and $\sin\alpha=\frac{\sqrt{2}}{3}$, find the value of $f\left(2016+\sin(\alpha-2\pi)\sin(\pi+\alpha)-2\cos^2(-\alpha)\right)$.

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