Let $U$ be a finite universal set, and let $A, B \subseteq U$ be two fixed subsets. We seek the number of subsets $X \subseteq U$ that satisfy the following intersection equation involving symmetric differences: $(A \Delta X) \cap (B \Delta X) = \emptyset$ If $|U| = 12$, $|A| = 7$, $|B| = 6$, and $|A \cap B| = 3$, how many such subsets $X$ exist?