
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: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$), the distance from its right focus to the line $x - y + 3\sqrt{2} = 0$ is 5, and the distance between one endpoint of the major axis and one endpoint of the minor axis of ellipse C is $\sqrt{10}$.
(1) Find the standard equation of ellipse C;
(2) Given a fixed point Q ($\frac{6\sqrt{5}}{6}$, 0), for any chord AB of ellipse C passing through Q, is $\frac{1}{|QA|^{2}} + \frac{1}{|QB|^{2}}$ a constant? If so, find the constant; if not, explain why.

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