
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 a polar coordinate system with the origin as the pole and the positive half of the $x$-axis as the polar axis, the polar equation of curve $C_{1}$ is $\rho=2$. Rectangle $ABCD$ is inscribed in curve $C_{1}$, with points $A$ and $B$ having polar coordinates $(2, \frac{\pi}{6})$ and $(2, \frac{5\pi}{6})$, respectively. By keeping the $x$-coordinates of all points on curve $C_{1}$ unchanged and halving their $y$-coordinates, curve $C_{2}$ is obtained.  
$(1)$ Write the Cartesian coordinates of $C$, $D$ and the parametric equation of curve $C_{2}$.  
$(2)$ Let $M$ be any point on curve $C_{2}$. Find the range of values for $|MA|^{2}+|MB|^{2}+|MC|^{2}+|MD|^{2}$.

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