
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 vectors $\vec{m}=(\sin x,\sqrt{3}\sin x)$, $\vec{n}=(\sin x,-\cos x)$, let the function $f(x)=\vec{m}\cdot \vec{n}$.    

$(1)$ Find the intervals of monotonic increase for the function $f(x)$ on $\left[ 0, \frac{3\pi}{2} \right]$;

$(2)$ In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, with $A$ being an acute angle. If $f(A)+\sin \left( 2A- \frac{\pi}{6} \right)=1$, $b+c=7$, and the area of $\triangle ABC$ is $2 \sqrt{3}$, find the length of side $a$.

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