Given a set $S$ of integers and four nonzero integer coefficients $\alpha_1$, $\alpha_2$, $\alpha_3$, $\alpha_4$, and a target $t$ such that $\gcd(\alpha_1, \alpha_2, \alpha_3)$ divides $t$, determine whether there exists four distinct $x_1, x_2, x_3, x_4\in X$ such that $\sum_{i=1}^4 \alpha_i x_i = t$.