
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 $\overrightarrow{a}=(2\sin x,\sqrt{3}\cos x)$ and $\overrightarrow{b}=(-\sin x,2\sin x)$, and the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$:

(I) Find the interval(s) where $f(x)$ is monotonically increasing;

(II) In $\triangle ABC$, $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively. Given that $f(C)=1$, $c=1$, $ab=2\sqrt{3}$, and $a > b$, find the values of $a$ and $b$.

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