
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
Selected Exercise $(4-4)$: Coordinate Systems and Parametric Equations

Given that the parametric equation of line $l$ is $\begin{cases} & x=-\frac{\sqrt{2}}{2}t \\ & y=\frac{\sqrt{2}}{2}t \\ \end{cases}$ ($t$ is the parameter), establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar coordinate equation of circle $C$ is $\rho=2\cos\theta$.

$(1)$ Find the ordinary equation of line $l$ and the rectangular coordinate equation of circle $C$.

$(2)$ Points $P$ and $Q$ move on line $l$ and circle $C$, respectively. Find the minimum value of $|PQ|$.

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