
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 an isosceles trapezoid with $\overline{AD}||\overline{BC}$ where the angle at the longer base $\overline{AD}$ is $\dfrac{\pi}{4}$. The diagonals are both $20\sqrt{10}$ long, and point $E$ is at distances $20\sqrt{5}$ and $40\sqrt{5}$ from vertices $A$ and $D$, respectively. Let $F$ be the foot of the altitude from $C$ to $\overline{AD}$. Find the distance $EF$, which can be expressed as $m\sqrt{n}$, where $m$ and $n$ are integers, and $n$ is square-free.

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