
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]


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In the Cartesian coordinate system $XOY$, the parametric equation of curve $C_{1}$ is $\begin{cases} x = 2\cos \varnothing \\ y = \sin \varnothing \end{cases} (\varnothing \text{ is the parameter})$, and in the polar coordinate system with $O$ as the pole and the positive half-axis of $x$ as the polar axis, curve $C_{2}$ is a circle with center at $(3,\frac{\pi}{2})$ and radius $1$.

$(1)$ Find the general equation of curve $C_{1}$ and the Cartesian coordinate equation of curve $C_{2}$;

$(2)$ Let $M$ be a point on curve $C_{1}$ and $N$ be a point on curve $C_{2}$. Find the range of values for $|MN|$.

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