
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

In the Cartesian coordinate system $xOy$, the curve $C_1$ is given by the parametric equations $\begin{cases} & x=2\cos \varphi \\ & y=\sin \varphi \end{cases}$ (with $\varphi$ as the parameter), and in the polar coordinate system with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the curve $C_2$ is a circle with center at $(3,\frac{\pi}{2})$ and radius $1$.

$(1)$ Find the standard 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