
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 $\overrightarrow{a}=(2\cos x, \sqrt{3}\cos x)$ and $\overrightarrow{b}=(\cos x, 2\sin x)$, where $x\in \mathbb{R}$, let $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$.  
$(Ⅰ)$ Find the smallest positive period of $f(x)$.  
$(Ⅱ)$ Find the maximum and minimum values of $f(x)$ on the interval $[0, \frac{\pi}{2}]$.

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