Find the number of positive integers \(n\) such that there exists an integer \(m \neq \frac{n}{2}\) such that \(\frac{n^3-2m^3}{6m}\) is a perfect square.

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**Details and Assumptions:**

- All perfect squares are integers. For example, \(8^2, 4^2,\) etc. are perfect squares, but \(\left( \frac{2}{3} \right) ^2\) isn't.

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