The idea revolves around initiating the process by first addressing $S_1$, followed by the intersection $S_1 \cap S_2$. This approach not only streamlines the computation but also potentially reduces the complexity involved when moving onto subsequent sets and their intersections, such as $S_2$, $S_1 \cap S_2 \cap S_3$, $S_1 \cap S_3$, $S_2 \cap S_3$, and finally $S_3$.

The proposed methodology suggests a structured way to handle sets and their intersections that could minimize redundant calculations and enhance the efficiency of the overall process. By focusing on intersections at the outset, there's an inherent optimization in managing the computational load and possibly improving the algorithm's performance by reducing the amount of data being processed in each step.

Moreover, this strategy might offer a clearer framework for understanding the relationships between different sets and their intersections, making it easier to visualize and implement the operations needed. In essence, prioritizing intersections could serve as a foundational tactic in optimizing set operations, potentially leading to more elegant and efficient solutions in programming tasks that involve complex set manipulations.