
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 the ellipse $5x^{2}+9y^{2}=45$, with its right focus point $F$,
(1) Find the length of the chord intercepted by the ellipse from the line passing through point $F$ with a slope of $1$;
(2) Determine the relationship between point $A(1,1)$ and the ellipse, and find the equation of the line containing the chord with $A$ as its midpoint.

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