For sets $A, B, X$ satisfying: $A \cap X = B \cap X = A \cap B, A \cup B \cup X = A \cup B$. If $A, B$ are known sets, find the set $X$.

If sets $A, B$ are known, determine the set $X$ based on the conditions provided.
In the study of operations between sets, it is important to understand the meaning of set operations and pay attention to the properties of the operations.

Solution one: From $A \cup B \cup X=A \cup B \Rightarrow X \subseteq A \cup B$. Let $x \in X \Rightarrow x \in A \cup B \Rightarrow$ $x \in A$ or $x \in B$. Since $x \in X$, we get $\left\{\begin{array}{l}x \in A \\ x \in X\end{array}\right.$ or $\left\{\begin{array}{l}x \in B \\ x \in X,\end{array}\right.$ which means $x \in(A \cap X) \cup(B \cap$ $X)=A \cap B$. From $x \in X \Rightarrow x \in A \cap B$, we get $X \subseteq A \cap B$. Also, $A \cap X=A \cap B \Rightarrow$ $A \cap B \subseteq X$. Therefore, $X=A \cap B$.

Solution two: From $A \cup B \cup X=A \cup B \Rightarrow X \subseteq A \cup B$, so $X=X \cap(A \cup B)=$ $(A \cap X) \cup(B \cap X)=A \cap B$.