
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 vectors $\overrightarrow{a} = (2\cos^2 x, \sqrt{3})$ and $\overrightarrow{b} = (1, \sin 2x)$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} - 2$.
(I) Find the minimum value of the function $f(x)$ on the interval $\left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$.
(II) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. If $f(C) = 1$, $c = 1$, $ab = 2\sqrt{3}$, and $a > b$, find the values of sides $a$ and $b$.

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