Finite type multiple flag varieties of exceptional groups

Consider a simple complex Lie group $G$ acting diagonally on a triple flag variety $G\mathrm{/}{P}_1\times G\mathrm{/}{P}_2\times G\mathrm{/}{P}_3$, where $P_i$ is parabolic subgroup of $G$. We provide an algorithm for systematically checking when this action has finitely many orbits. We then use this method to give a complete classification for when $G$ is of type $F_4$. The $E_6,E_7,$ and $E_8$ cases will be treated in a subsequent paper.