The incircle of $\triangle ABC$ is tangent to sides $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ at $D$, $E$, and $F$, respectively. Point $G$ is the intersection of lines $AC$ and $DF$. The sides of $\triangle ABC$ have lengths $AB = 73$, $BC = 123$, and $AC = 120$. Find the length $EG$.

Let's think step by step and output the final answer within \boxed{}.
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