
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 the polar equation of circle $C$ is $\rho =2\cos \theta$, and the parametric equation of line $l$ is:

\[
\begin{cases}
x= \frac{1}{2}+ \frac{ \sqrt{3}}{2}t \\
y= \frac{1}{2}+ \frac{1}{2}t
\end{cases} \quad (t\text{ is the parameter}),
\]

Suppose line $l$ intersects circle $C$ at points $P$ and $Q$, and point $A$ is a point on line $l$ different from $P$ and $Q$.

(Ⅰ) Find the Cartesian coordinate equation of circle $C$;

(Ⅱ) Find the value of $|AP|\cdot|AQ|$.

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