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

$(1)$ When $\sin \theta = -\frac{1}{2}$, find the maximum and minimum values of $f(x)$;

$(2)$ If $f(x)$ is a monotonic function on $x \in \left[ -\frac{1}{2}, \frac{\sqrt{3}}{2} \right]$ and $\theta \in \left[ 0, 2\pi \right)$, find the range of values for $\theta$.

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