Sets of Perfect Squares

Let \(\mathcal{S}\) be the set of all perfect squares whose rightmost three digits in base \(10\) are \(256\). Let \(\mathcal{T}\) be the set of all numbers of the form \(\frac{x^2-256}{1000}\), where \(x\) is in \(\mathcal{S}\). In other words, \(\mathcal{T}\) is the set of numbers that result when the last three digits of each number in \(\mathcal{S}\) are truncated. Find the remainder when the tenth smallest element of \(\mathcal{T}\) is divided by \(1000\).

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