Five five five and five!

Algebra Level 4

Let \(a, b, c, d, e \in R^{+}\), their sum is equal to 8 and their sum of squares is equal to 16. If the maximum value of \(e\) can be written as \(\frac{A}{B}\), when \(A\) and \(B\) are coprime positive integers. Find \(A + B\).

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