
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 $\overrightarrow{a}=(\cos x,\cos x)$, $\overrightarrow{b}=(\sin x,-\cos x)$, let the function $f(x)=2\overrightarrow{a}\cdot\overrightarrow{b}+1$, where $x\in\mathbb{R}$.  
(Ⅰ) Find the smallest positive period of the function $f(x)$ and the coordinates of the symmetry center of the graph of the function $f(x)$.  
(Ⅱ) If $\alpha\in(0,\frac{\pi}{2})$ and $f(\frac{\alpha}{2})=\frac{2}{3}$, find the value of $\cos 2\alpha$.

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