Obviously we must have $a,b,c\equiv 1\text{(mod 2)}$. But then the product of the first terms is $(abc)^4\equiv 1\text{(mod 4)}$, so it is impossible for all three to be $3\text{(mod 4)}$, and thus one of the numbers will be $3\text{(mod 4)}$, so the answer is $\boxed{\text{No}}$.