
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
Elective 4-4: Coordinate System and Parametric Equations

It is known that in the Cartesian coordinate system, the parametric equation of curve $C$ is $\begin{cases} & x=1+2\cos \varphi \\ & y=1+2\sin \varphi \\ \end{cases}$ ($\varphi$ is the parameter), and now establishing a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of line $l$ is $\sqrt{2}p\cos (\theta +\frac{\pi }{4})=4$.

$(1)$ Find the general equation of curve $C$ and the Cartesian coordinate equation of line $l$;

$(2)$ If point $P$ is any point on curve $C$, find the minimum distance from point $P$ to line $l$.

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