
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 $\Delta ABC$, where $A$, $B$, and $C$ are opposite to sides $a$, $b$, and $c$ respectively, given $f\left( \frac{A}{2}\right)=0, a=1$, find the maximum area of $\Delta ABC$.

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