
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 $\vec{n} = (\sqrt{3} \sin x,2)$, $\vec{n} = (2\cos x,\cos^2 x)$, $f(x)= \vec{m} \cdot \vec{n}$.

$(1)$ Find the analytical expression and the smallest positive period of $f(x)$.

$(2)$ Find the intervals where $f(x)$ is monotonically increasing.

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