Given three sets of integers $A_1, A_2, A_3$, three nonzero integer coefficients $\alpha_1$, $\alpha_2$, $\alpha_3$, and a target $t$ such that $\gcd(\alpha_1, \alpha_2, \alpha_3)$ divides $t$, determine whether there exists $x_1\in A_1$, $x_2\in A_2$, $x_3\in A_3$ such that $\sum_{i=1}^3 \alpha_i x_i = t$.