
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 $f(x)=\sin^2x+ \sqrt{3}\tan \theta \cdot \cos x+ \frac{ \sqrt{3}}{8}\tan \theta - \frac{3}{2}$, where $\theta \in [0, \frac{\pi}{2}]$ and $\theta \in [0, \frac{\pi}{3}]$

$(1)$ If $\theta= \frac{\pi}{3}$, find the maximum value of $f(x)$ and the corresponding value of $x$;

$(2)$ Is there a real number $\theta$ such that the maximum value of $f(x)$ is $- \frac{1}{8}$? If it exists, find the corresponding value of $\theta$; if not, explain why.

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