
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 $f(x)=2 \sqrt {3}\cos ^{2}x+2\sin x\cos x-m$ $(x\in\mathbb{R})$, the maximum value of the function $f(x)$ is $2$.  
$(1)$ Find the value of the real number $m$;  
$(2)$ In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $A$ is an acute angle, and it satisfies $f(A)=0$, $\sin B=3\sin C$, and the area of $\triangle ABC$ is $\frac {3 \sqrt {3}}{4}$, find the length of side $a$.

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