Let $a_{ij}$, $1 \leq i,j \leq 3$, $b_1, b_2, b_3$, and $c_1, c_2, c_3$ be positive real numbers. Let $S$ be the set of triples of positive real numbers $(x, y, z)$ such that: \[ a_{11}x + a_{12}y + a_{13}z \leq b_1, \quad a_{21}x + a_{22}y + a_{23}z \leq b_2, \quad a_{31}x + a_{32}y + a_{33}z \leq b_3. \]Let $M$ be the largest possible value of $f(x, y, z) = c_1x + c_2y + c_3z$ for $(x, y, z) \in S$. Let $T$ be the set of triples $(x_0, y_0, z_0)$ from $S$ such that $f(x_0, y_0, z_0) = M$. Prove that if $T$ contains at least two distinct triples, then $T$ is an infinite set.