
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 $\vec{a} = (5\sqrt{3}\cos{x}, \cos{x})$, $\vec{b} = (\sin{x}, 2\cos{x})$, let the function $f(x) = \vec{a} \cdot \vec{b} + |\vec{b}|^{2} + \frac{3}{2}$.
(1) Find the smallest positive period and the center of symmetry for the function $f(x)$;
(2) When $x \in \left[ \frac{\pi}{6}, \frac{\pi}{2} \right]$, find the range of the function $f(x)$.

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