
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 $\overrightarrow{m}=(2\cos ^{2}x, \sqrt {3})$ and $\overrightarrow{n}=(1,\sin 2x)$, and the function $f(x)= \overrightarrow{m} \cdot \overrightarrow{n}$.
(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. Given $f(C)=3$, $c=1$, $ab=2 \sqrt {3}$, and $a > b$, find the values of $a$ and $b$.

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