
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 $f(x)=2\cos x\cdot \sin \left(x+ \frac{\pi }{6}\right)+\sqrt{3}\sin x\cdot \cos x-\sin ^{2}x$,

(1) Find the monotonically increasing interval of the function $y=f(x)$ ($0 < x < \pi $);

(2) Let $\Delta ABC$'s internal angle $A$ satisfy $f\left(A\right)=2$, and $\overset{→}{AB}\cdot \overset{→}{AC}= \sqrt{3}$, find the maximum length of the height $AD$ on edge $BC$.

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