Given that \(x^2 + y^2 = 4\) , solve for:

\[\large {\max(\lfloor\sqrt{3x}\rfloor + \lfloor\sqrt{4y}\rfloor + \lfloor\sqrt[4]{3x}\rfloor + \lfloor\sqrt[4]{4y}\rfloor + \lfloor\sqrt[8]{3x}\rfloor + \lfloor\sqrt[8]{4y}\rfloor + \lfloor\sqrt[16]{3x}\rfloor + \lfloor\sqrt[16]{4y}\rfloor + \cdots + \lfloor\sqrt[512]{3x}\rfloor + \lfloor\sqrt[512]{4y}\rfloor)} \]

For more problems like this, try answering this set .

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