Example 4 Given three sets of real numbers $A=\left\{x \mid x^{2}-3 x+2=0\right\}, B=\left\{x \mid x^{2}-a x+(a-1)=0\right\}, C=\left\{x \mid x^{2}-m x+2=0\right\}$, if $A \cup B=A, A \cap C=C$, try to find the values of $a$ and $m$.

---

The translation maintains the original text's formatting and line breaks.
Given the problem, $A=\{1,2\}, B=\{x \mid[x-(a-1)](x-1)=0\}$, from $A \cup B=$ $A, A \cap C=C$ we know, $B \subseteq A, C \subseteq A$, so $a=2$ or $3 ; m=3$ or $-2 \sqrt{2}<m<2 \sqrt{2}$.

Key to solving the problem: Using set language to express mathematical concepts or relationships is concise and precise, but it can also make understanding the problem more difficult. Typically, the following are equivalent expressions: $A \cup B=A \Leftrightarrow B \subseteq A \Leftrightarrow A \cap B=B \Leftrightarrow B$ $\subseteq A \cap B \Leftrightarrow A \cup B \subseteq A$