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

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

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

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