
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 \( \overline{AD} \parallel \overline{BC} \) and bases \(AD\) longer than \(BC\). The angle at the longer base \( \overline{AD} \) is \( \frac{\pi}{4} \). The diagonals have length \( 12\sqrt{15} \), and point \( E \) is at distances \( 12\sqrt{5} \) and \( 36\sqrt{5} \) from vertices \( A \) and \( D \), respectively. Let \( G \) be the midpoint of \( \overline{BC} \). Determine the distance \( EG \) which 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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