
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 $\overline{AD}||\overline{BC}$ and angles at the larger base $\overline{AD}$ equal to $\frac{\pi}{4}$. Suppose the diagonals each have length $20\sqrt{2}$, and point $E$ is at distances $20$ and $40$ from vertices $A$ and $D$, respectively. Let $F$ be the foot of the altitude from $B$ to $\overline{AD}$. Determine the distance $EF$ and express it in the form $m\sqrt{n}$, with $m$ and $n$ positive integers, and $n$ is not divisible by the square of any prime.

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