
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with its right focus having a distance of 5 to the line $x - y + 3\sqrt{2} = 0$, and the distance between one endpoint of the major axis and one endpoint of the minor axis is $\sqrt{10}$.

1. Find the equation of the ellipse $C$.
2. Determine if there exists a point $Q$ on the $x$-axis such that the line passing through $Q$ intersects the ellipse $C$ at points $A$ and $B$, satisfying $\frac{1}{QA^{2}}+ \frac{1}{QB^{2}}$ as a constant value. If such a point exists, find the constant value and the coordinates of point $Q$. If not, provide a reason.

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