Prove or give a counterexample:

For three positive integers $\displaystyle u_1<u_2<u_3$, if $\displaystyle gcd(u_2,u_3)=gcd(u_1,u_3)=gcd(u_1,u_2)=1$ then $\displaystyle gcd(u_1u_2u_3,u_2u_3+u_1u_3+u_1u_2)=1$

Note: This is stronger than the condition that $\displaystyle gcd(u_1,u_2,u_3)=1$, since examples like $\displaystyle (1,3,6)$ prove false.