Calling All Powers That Give Two Digits

Let \(A,B,C\) and \(D\) be distinct single digit positive integers such that: \[ A^B =\overline{CD} \text{ and } A+B=C+D \]

Compute \( \overline{AB} + \overline{BC} + \overline{CD} + \overline{DA} \).

Clarification: \(\overline{AB}\) denotes a 2-digit integer with \(A\) and \(B\) as its digits.

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