
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 $f(x)= \vec{a} \cdot \vec{b}$, where $\vec{a}=(2\cos x, -\sqrt{3}\sin 2x)$ and $\vec{b}=(\cos x, 1)$, with $x\in \mathbb{R}$.

 $(1)$ Find the interval of monotonic increase for $f(x)$;

 $(2)$ In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, $f(A)=-1$, $a= \frac{\sqrt{7}}{2}$, and vectors $\vec{m}=(3,\sin B)$ and $\vec{n}=(2,\sin C)$ are collinear. Find the values of sides $b$ and $c$.

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