
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: $f(x) = \sqrt{3} \cos(2x) + 2 \sin(\frac{3}{2}\pi + x) \sin(\pi - x), x \in \mathbb{R}$
(1) Find the smallest positive period and the monotonically increasing interval of $f(x)$.
(2) In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given $f(A) = -\sqrt{3}$ and $a = 3$, find the maximum value of the altitude on side $BC$.

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