
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 $f(x)=\cos x\sin x- \sqrt {3}\cos ^{2}x+ \dfrac { \sqrt {3}}{2}$.
$(1)$ Find the intervals of increase for $f(x)$;
$(2)$ In $\triangle ABC$, $A$ is an acute angle and $f(A)= \dfrac { \sqrt {3}}{2}$, $D$ is the midpoint of $BC$, $AD=3$, $AB= \sqrt {3}$, find the length of $AC$.

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