HMMT 2022 G9

Let $A_{1} B_{1} C_{1}, A_{2} B_{2} C_{2}$, and $A_{3} B_{3} C_{3}$ be three triangles in the plane. For $1 \leq i \leq 3$, let $D_{i}, E_{i}$, and $F_{i}$ be the midpoints of $B_{i} C_{i}, A_{i} C_{i}$, and $A_{i} B_{i}$, respectively. Furthermore, for $1 \leq i \leq 3$ let $G_{i}$ be the centroid of $A_{i} B_{i} C_{i}$. Suppose that the areas of the triangles $A_{1} A_{2} A_{3}, B_{1} B_{2} B_{3}, C_{1} C_{2} C_{3}, D_{1} D_{2} D_{3}, E_{1} E_{2} E_{3}$, and $F_{1} F_{2} F_{3}$ are $2,3,4,20,21$, and 2020, respectively. Compute the largest possible area of $G_{1} G_{2} G_{3}$.