
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 $f(x)=x^{2}+2x\cdot\tan \theta-1$, where $x\in[-1, \sqrt{3}]$ and $\theta\in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

$(1)$ When $\theta=-\frac{\pi}{6}$, find the maximum and minimum values of the function $f(x)$;  
$(2)$ Find the range of $\theta$ such that $y=f(x)$ is a monotonic function on the interval $[-1, \sqrt{3}]$.

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