
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
In the plane rectangular coordinate system $xOy$, the parameter equation of circle $C$ is $\left\{\begin{array}{l}x=rcosα\\ y=rsinα\end{array}\right.$ ($\alpha$ is the parameter, $r \gt 0$), with the coordinate origin $O$ as the pole, and the positive half-axis of the $x$-axis as the polar axis to establish a polar coordinate system. The polar equation of the line $l$ is $\sqrt{2}\rho \cos (\theta +\frac{π}{4})=4$. <br/>$(1)$ Find the general equation of circle $C$ and the Cartesian equation of line $l$, and determine the value of $r$ when the line $l$ is tangent to circle $C; <br/>$(2)$ Points $P$ and $Q$ are on the line $l$ and circle $C$ respectively, if $r=\sqrt{2}$, find the minimum value of the length of line segment $PQ$.

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