
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 $f(x) = 1 + 2\sqrt{3}\sin x\cos x - 2\sin^2 x$, where $x \in \mathbb{R}$.

$(1)$ Find the intervals of monotonicity for the function $f(x)$;

$(2)$ If the function $f(x)$ is translated to the right by $\frac{\pi}{6}$ units to obtain the function $g(x)$, find the minimum and maximum values of $g(x)$ on the interval $\left[-\frac{\pi}{2}, 0\right]$.

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