For positive real numbers \(x\) and \(y\), define their *special mean* to be the average of their arithmetic and geometric means. Find the total number of pairs of integers \((x,y)\), with \(x \le y\), from the set of numbers \(\{1,2,3...,2016\}\), such that the special mean of \(x\) and \(y\) is a perfect square.

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