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$y = \pm1$: $x^2 + 4(1) = 16 \Rightarrow x^2 = 12$ (no integer solutions).
$y = \pm2$: $x^2 + 4(4) = 16 \Rightarrow x^2 = 0 \Rightarrow x = 0$ (2 points).
Total lattice points: $2 + 2 = \boxed{4}$.

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