
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 $xOy$, there is a curve $C_{1}: \begin{cases} x=3\cos \theta \\ y=2\sin \theta \end{cases}$ (with $\theta$ as the parameter). In the polar coordinate system with $O$ as the pole and the positive half-axis of $x$ as the polar axis, there is a curve $C_{2}: \rho-2\cos \theta=0$.
$(1)$ Find the Cartesian equation of curve $C_{2}$;
$(2)$ If there is a moving point $M$ on curve $C_{1}$ and a moving point $N$ on curve $C_{2}$, find the minimum value of $|MN|$.

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