An inscribed Hexagon has sides \( AF = FE = ED = 4 \), \( AB = BC = CD = 2 \). Furthermore, \(ADEF\) and \(ADCB\) are trapeziums.

If the length of chord \(AD\) is \( \frac{ m}{n} \), where \(m\) and \(n\) are relatively prime integers, find \(m + n \).

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