Complex on Complex

Algebra Level 5

Suppose that:

\[\large \left(\sqrt{2} + i\dfrac{\sqrt{2}}{3}\right)^{(1+i)} = \alpha + i \beta\]

where, \(\alpha\) and \(\beta\) are real numbers and \(i^2 = - 1\). Find \(\lfloor 1000\alpha \rfloor + \lfloor 1000\beta \rfloor\), where \(\lfloor x \rfloor\) is the greatest integer function.

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