
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 $f(x)=\frac{1}{2}\sin\omega x-\frac{\sqrt{3}}{2}\sin(\omega x+\frac{\pi}{2})$ where $(\omega > 0)$ has exactly $3$ zeros in the interval $[0,\frac{3\pi}{2}]$, where $\omega$ is a positive integer.  
$(1)$ Find the analytical expression of the function $f(x)$;  
$(2)$ The graph of the function $f(x)$ is shifted to the left by $\frac{\pi}{4}$ units to obtain the graph of the function $g(x)$. If $g(\frac{\alpha}{2})=\frac{4}{5}$, find the value of $g(-\alpha)$.

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