
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 $f(x)=2\cos^2x+\sin\left(\frac{7\pi}{6}-2x\right)-1$ ($x\in\mathbb{R}$).

$(1)$ Find the smallest positive period of the function $f(x)$ and the intervals of monotonic increase;

$(2)$ In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $f(A)=\frac{1}{2}$, if $b+c=2a$ and $\overrightarrow{AB}\cdot\overrightarrow{AC}=6$, find the value of $a$.

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