Exceptional groups and the $s$-arc-transitivity of vertex-primitive digraphs, I

In this paper, we study the primitive actions of almost simple exceptional groups of Lie type on $s$-arc-transitive digraphs. Our motivation is the following question posed by Giudici and Xia: Is there an upper bound on $s$ for finite vertex-primitive $s$-arc-transitive digraphs that are not directed cycles? In a 2018 paper, Giudici and Xia reduced this question to the case where the automorphism group of the digraph is an almost simple group with socle $L$. Subsequently, it has been proved that $s\le 2$ when $L$ is a linear, symplectic or alternating group, and $s\le 1$ when $L$ is a Suzuki group, a small Ree group, or one of $22$ specific sporadic groups. In this paper, we prove that $s\le 2$ when $L$ is ${}^3{D}_4(q)$, $G_2(q)$ (including $G_2(2{)}^{\mathrm{\prime }}$), ${}^2{F}_4(q)$ (including ${}^2{F}_4(2{)}^{\mathrm{\prime }}$), $F_4(q)$, $E_6(q)$ or ${}^2{E}_6(q)$.