
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 the function $f(x)=2\sin(\pi-x)\cos(x)-2\sqrt{3}\cos^2(x)$.
$(1)$ Find the maximum value of $f(x)$.
$(2)$ In acute triangle $\triangle ABC$, where $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, if $f(A)=0$, $b=4$, and $c=3$, find the length of the altitude $AD$ from side $BC$.

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