Countably many

There are \(N\) rational numbers \(R \) such that

\[ \sqrt[3] {5\times 10^8 + \sqrt{R} } + \sqrt[3]{5\times 10^8- \sqrt{R} } \]

is an integer. What are the last three digits of \(N\)?

Details and assumptions

You may use the fact that \( 0.7937 < \sqrt[3]{0.5} < 0.793701 \)

We choose the real value of the roots (where applicable).

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