
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 $f(x) = 2\cos^2 x + 2\sqrt{3}\sin x\cos x \ (x \in \mathbb{R})$.
(Ⅰ) When $x \in [0, \frac{\pi}{2}]$, find the interval where the function $f(x)$ is monotonically increasing.
(Ⅱ) Suppose $\triangle ABC$ has interior angles at $A$, $B$, $C$, and the sides opposite to these angles are $a$, $b$, $c$, respectively, with $c=3$, $f(C)=2$. If the vector $\overrightarrow{m} = (1, \sin A)$ is collinear with the vector $\overrightarrow{n} = (2, \sin B)$, find the values of $a$ and $b$.

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