
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
Let $ABCD$ be a rectangle with $\overline{AB}$ parallel to $\overline{CD}$ and longer than $\overline{BC}$. The diagonals of the rectangle have length $20\sqrt{5}$. Point $E$ is inside the rectangle at distances $8\sqrt{5}$ from vertex $A$ and $12\sqrt{5}$ from vertex $B$. Let $F$ be the midpoint of $\overline{CD}$. Compute the distance $EF$, which can be expressed as $x\sqrt{y}$, where $x$ and $y$ are positive integers with $y$ not divisible by the square of any prime. Find $x + y$.

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