Given this 'infinitely long' fractional equation: \( \sqrt{6} = A + \frac {1}{B + \frac {1}{C + \frac {1}{D + \frac {1}{E + \ldots}}}} \)

It is also given that the values of the consecutive variables A, B, C, ..., X, Y, and Z are positive integers.

Compute the value of: \( A+B+C+D+E+\ldots+X+Y+Z \)

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