Logarithm Sums

Algebra Level 3

\(x\) and \(y\) are positive real numbers that satisfy \(\log_x y + \log_y x = \frac{17}{4}\) and \(xy = 288\sqrt{3} \). If \(x + y = a + b\sqrt{c}\), where \(a\), \(b\) and \(c\) are positive integers and \(c\) is not divisible by the square of any prime, what is the value of \(a + b + c\)?

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