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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