
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
Given an ellipse $C$ with its center at the origin and the major axis along the $y$-axis, passing through points $M(4,2)$ and $N(\sqrt{6},3)$.<br/>$(1)$ Find the equation of the ellipse $C$;<br/>$(2)$ Let $R(x_{0},y_{0})$ be any point on the ellipse $C$. Two tangents are drawn from the origin $O$ to the circle $R:(x-x_{0})^{2}+(y-y_{0})^{2}=8$, intersecting the ellipse at $P$ and $Q$. Investigate whether $OP^{2}+OQ^{2}$ is a constant value. If it is, find its value; if not, explain the reason.

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