$(a,b,c)=(6t,6t+1,6t+5)$ where $t\equiv_{35} 1$ work

Prove that there exist infinitely many positive integer triples $(a,b,c,d)$ such that $a ,b,c,d$ are pairwise relatively prime ,and $abc+d ,abd+c ,acd+b,bcd+a$ are pairwise relatively prime .

For the original problem, $(p,p+1,p+2)$ for any prime $p>2$ also works. Conclusion follows easily from finding gcd.

For the original problem, $(2,3,30k+5)$ for $k\in \mathbb{N}$ works as well.