
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 vectors $\overrightarrow{a}=(5 \sqrt {3}\cos x,\cos x)$ and $\overrightarrow{b}=(\sin x, 2\cos x)$, let the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}+|\overrightarrow{b}|^{2}+\frac{3}{2}$.

1. Find the smallest positive period and the symmetry center of the function $f(x)$.
2. When $x\in [\frac{\pi}{6},\frac{\pi}{2}]$, find the range of the function $f(x)$.

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