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

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