
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 $f(x)= \sqrt{3}\sin (\omega x+\varphi)$ $(\omega > 0,- \frac{\pi}{2}\leqslant \varphi < \frac{\pi}{2})$ whose graph is symmetric about the line $x= \frac{\pi}{3}$, and the distance between two adjacent highest points on the graph is $\pi$.

$(I)$ Find the values of $\omega$ and $\varphi$;

$(II)$ If $f\left( \frac{\alpha}{2}\right)= \frac{ \sqrt{3}}{4}$ $\left( \frac{\pi}{6} < \alpha < \frac{2\pi}{3}\right)$, find the value of $\cos \left( \alpha +\frac{3\pi }{2} \right)$.

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