Logarithm Sums

Algebra Level 3

xx and yy are positive real numbers that satisfy logxy+logyx=174\log_x y + \log_y x = \frac{17}{4} and xy=2883xy = 288\sqrt{3} . If x+y=a+bcx + y = a + b\sqrt{c}, where aa, bb and cc are positive integers and cc is not divisible by the square of any prime, what is the value of a+b+ca + b + c?

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