
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 Cartesian coordinate system $xOy$, the parametric equation of curve $C_1$ is $\begin{cases}x=-3+t \\ y=-1-t\end{cases} (t$ is the parameter), with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of circle $C_2$ is $p=4\sqrt{2}\sin\left(\frac{3\pi}{4}-\theta\right)$.

$(1)$ Find the general equation of $C_1$ and the Cartesian coordinate equation of $C_2$;

$(2)$ Let point $M$ be any point on the curve $C_1$, and a tangent line to circle $C_2$ is drawn through $M$, with the tangent point being $N$. Find the minimum value of $|MN|$.

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