How many ordered pairs of non-negative integers \((a,b)\) subject to \(a,b <1000\) are there such that both \(a^2+4b\) and \(b^2+4a\) are perfect squares?

This problem is a reworded version of a problem that appeared in APMO-1999.

This problem is a part of the set "Olympiads and Contests Around the World - 2". You can see rest of the problems here.

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