Statistics and Probability: JEE Advanced 2021 Mathematics question

Consider three sets $E_{1}=\{1,2,3\}, F_{1}=\{1,3,4\}$ and $G_{1}=\{2,3,4,5\}$. Two elements are chosen at random, without replacement, from the set $E_{1}$, and let $S_{1}$ denote the set of these chosen elements. Let $E_{2}=E_{1}-S_{1}$ and $F_{2}=F_{1} \cup S_{1}$. Now two elements are chosen at random, without replacement, from the set $F_{2}$ and let $S_{2}$ denote the set of these chosen elements. Let $G_{2}=G_{1} \cup S_{2}$. Finally, two elements are chosen at random, without replacement, from the set $G_{2}$ and let $S_{3}$ denote the set of these chosen elements. Let $E_{3}=E_{2} \cup S_{3}$. Given that $E_{1}=E_{3}$, let $p$ be the conditional probability of the event $S_{1}=\{1,2\}$. Then the value of $p$ is