
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 that the center of an ellipse is at the origin $O$, its minor axis has a length of $2\sqrt{2}$, one focus is at $F(c,0)$ $(c > 0)$, and one fixed point is at $A(\frac{10-c^2}{c},0)$, with $\overset{⇀}{OF}=2 \overset{⇀}{FA}$. The line passing through point $A$ intersects the ellipse at two points $P$ and $Q$:

(1) Determine the equation of the ellipse and its eccentricity.

(2) If a circle, passing through the origin, has $PQ$ as its diameter, determine the equation of line $PQ$.

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