The sides $x$ and $y$ of a scalene triangle satisfy the equation $x + \frac{2\Delta}{x} = y + \frac{2\Delta}{y}$, where $\Delta$ is the area of the triangle. Given that $x = 60$ and $y = 63$, determine the length of the largest side of the triangle.

Let's think step by step and output the final answer within \boxed{}.
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