
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 bases $\overline{AB}$ and $\overline{CD}$ where $\overline{AB}||\overline{CD}$, and the angle at the longer base $\overline{CD}$ is $\frac{\pi}{3}$. The diagonals have length $8\sqrt{21}$, and point $E$ is at distances $8\sqrt{7}$ and $24\sqrt{7}$ from vertices $C$ and $D$, respectively. Let $F$ be the foot of the altitude from $B$ to $\overline{CD}$. The distance $EF$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$.

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