
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 function $f(x)=2\sin x\cos x-2\sin ^{2}x+1$, in triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, where $a=\sqrt{3}$, $A$ is an acute angle, and $f(A+\frac{\pi }{8})=\frac{\sqrt{2}}{3}$. Find the maximum value of the area of triangle $ABC$.

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