432 Is Special

Let \( S \) be the set of rational numbers \( x \) such that \( x^3-432 \) is the square of a rational number.

If \( S \) is infinite, enter \( -1 \). If \( S \) is finite, enter the sum of the elements of \( S. \)

Here is a very helpful hint:
If \( y^2=x^3-432,\) let \( u = \dfrac{36-y}{6x}, \, v = \dfrac{36+y}{6x} \). What is \( u^3+v^3\)?

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