
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 $f(x) = \vec{a} \cdot \vec{b}$, where vectors $\vec{a} = (2\cos x, 1)$ and $\vec{b} = (\cos x, \sqrt{3} \sin 2x) (x \in \mathbb{R})$.
1. Find the smallest positive period of $f(x)$.
2. In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. If $f(A) = 2$, $a = \sqrt{3}$, $b + c = 3$, and $b > c$, find the lengths of $b$ and $c$.

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