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

(I) If $f(x)$ is a monotonic function on the interval $x \in [-1, \sqrt{3}]$, find the range of $\theta$.

(II) If, when $\theta \in \left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$, $y = f(x)$ has a minimum value of $g(\theta)$ on $[-1, \sqrt{3}]$, find the expression for $g(\theta)$.

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