No. Consider $$F(0)=0,\ F(1)=2,\ F(2)=1,\ F(3)=3$$ $$F(4)=0,\ F(5)=2,\ F(6)=1,\ F(7)=3$$ where a number represents its base-2 expansion, e.g. 6 represents $(1,1,0)$.
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