
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 $f(x)= \sqrt{3}\sin \omega x+\cos (\omega x+ \frac{\pi}{3})+\cos (\omega x- \frac{\pi}{3})-1$ ($\omega > 0$, $x\in\mathbb{R}$),

and the smallest positive period of the function $f(x)$ is $\pi$.

$(1)$ Find the analytical expression of the function $f(x)$;

$(2)$ In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $\alpha$ are $l$, $\alpha$, and $l$ respectively. If $\alpha$, $(\vec{BA}\cdot \vec{BC}= \frac{3}{2})$, and $a+c=4$, find the value of $b$.

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