
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 $f(x)=\sin x+a\cos x$, whose graph passes through the point $(\frac{\pi}{3},0)$.
(1) Find the value of the real number $a$;
(2) If $g(x)=[f(x)]^{2}-2$, find the range of the function $g(x)$ when $x\in(\frac{\pi}{4},\frac{2\pi}{3})$;
(3) If $g(\frac{a}{2})=-\frac{\sqrt{3}}{4}$ $(\frac{\pi}{6} < a < \frac{2\pi}{3})$, find the value of $\cos(\alpha+\frac{3\pi}{2})$.

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