
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 $AB = CD = 30$ and $BC = DA = 40$. Point $E$ is inside the rectangle such that the distances from $E$ to $A$ and $B$ are $10$ and $30$, respectively. Let $\overline{EF}$ be perpendicular from $E$ to side $\overline{AD}$. Find the length of $\overline{EF}$, and express it in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime.

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