
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 $f(x) = \sin x \cos x - \cos^2\left(x + \frac{\pi}{4}\right), x \in \mathbb{R}$.  
(I) Find the intervals of monotonic increase for $f(x)$.  
(II) In an acute triangle $ABC$, where $A$, $B$, and $C$ are the opposite sides of $a$, $b$, and $c$ respectively, if $f\left(\frac{A}{2}\right) = 0$ and $a = 1$, find the maximum area of $\triangle ABC$.

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