
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 $f(x)=\sin x\cos x-\cos^{2}(x+\dfrac{\pi}{4})$, $x\in R$.

(1) Find the interval(s) where $f(x)$ is monotonically increasing.

(2) In acute triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $f(\dfrac{A}{2})=0$, $a=1$, find the maximum area of triangle $ABC$.

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