Given a set $S$ of integers, 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 three distinct $x_1, x_2, x_3\in X$ such that $\sum_{i=1}^3 \alpha_i x_i = t$.