
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
Let $f(x)=\sin(\omega x+\frac{\pi}{6})+2\sin^2(\frac{\omega x}{2})$ $(\omega>0)$ be a function. It is known that the distance between adjacent pairs of symmetry axes of the graph of the function $f(x)$ is $\pi$.  
$(1)$ Find the analytical expression of the function $f(x)$.  
$(2)$ If in triangle $ABC$, the sides opposite to the angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively $(b < a)$, and $f(A)=\frac{3}{2}$, the area of $\triangle ABC$ is $S=6\sqrt{3}$, and $a=2\sqrt{7}$, find the values of $b$ and $c$.

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