
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 $f(x) = \sin x \cos x - \cos^2 \left( x + \frac{\pi}{4} \right)$.

(1) Find the intervals where $f(x)$ is monotonically increasing or decreasing.

(2) In an acute triangle $\Delta ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. If $f\left( \frac{A}{2} \right) = 0$ and $a = 1$, find the maximum area of $\Delta ABC$.

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