
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 with its center at the origin $O$, its minor axis is $2\sqrt{2}$, one of its foci is $F$ with coordinates $(c, 0)$ ($c>0$), and a fixed point $A$ with coordinates $\left(\frac{10}{c}-c, 0\right)$. It is also given that $\overrightarrow{OF} = 2\overrightarrow{FA}$. A line passing through point $A$ intersects the ellipse at points $P$ and $Q$:  
(1) Find the equation of the ellipse and its eccentricity;  
(2) If $OP \perp OQ$, find the equation of line $PQ$.

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