
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 $f(x) = \vec{m} \cdot \vec{n}$, where vectors $\vec{m} = (1, 2\cos x)$ and $\vec{n} = (\sqrt{3}\sin 2x, \cos x)$.
(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 that $f(A) = 2$, $b = 1$, and the area of $\triangle ABC$ is $\sqrt{3}$, find the radius $R$ of the circumcircle of $\triangle ABC$.

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