
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 $f(x) = 2\sin x \cos x - 2\sin^2 x + 1 (x \in \mathbb{R})$.

1. Find the smallest positive period and the monotonically increasing interval of the function $f(x)$.
2. In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $a = \sqrt{3}$, $A$ being an acute angle, and $f(A + \frac{\pi}{8}) = \frac{\sqrt{2}}{3}$. Find the maximum value of the area $S$ of $\triangle ABC$.

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