
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 that the polar coordinate equation of circle $C$ is $ρ=2\cos θ$, the parametric equation of line $l$ is $\begin{cases}x= \frac{1}{2}+ \frac{ \sqrt{3}}{2}t \\ y= \frac{1}{2}+ \frac{1}{2}t\end{cases} (t\text{ is a parameter})$, and the polar coordinates of point $A$ are $(\frac{ \sqrt{2}}{2} ,\frac{π}{4} )$, let line $l$ intersect with circle $C$ at points $P$ and $Q$.

(1) Write the Cartesian coordinate equation of circle $C$;

(2) Find the value of $|AP|⋅|AQ|$.

By Vieta's theorem, we get $t_{1}⋅t_{2}=- \frac{1}{2} < 0$, and according to the geometric meaning of the parameters, we have $|AP|⋅|AQ|=|t_{1}⋅t_{2}|= \frac{1}{2}$.

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