
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 $\overrightarrow{m} = (2\cos^2x, \sin x)$ and $\overrightarrow{n} = (1, 2\cos x)$.  
(I) If $\overrightarrow{m} \perp \overrightarrow{n}$ and $0 < x < \pi$, find the value of $x$;  
(II) Let $f(x) = \overrightarrow{m} \cdot \overrightarrow{n}$, find the equation of the axis of symmetry and the center of symmetry of $f(x)$.

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