Let $r \geq 2023$ be a rational number. The real numbers $a, b$, and $c$ satisfy \[ 4a^2 + 4b^2 + 9c^2 = r. \]Does there exist a value of $r$ for which the number of rational triples $(a,b,c)$ that achieve the maximum possible value of $4ab + 6bc - 6ac$ is: a) zero b) finite, but non-zero?