Example 12 Let $m>0, n>0$, and real numbers $a, b, c, d$ satisfy $a+b+c+d=m, ac=bd=n^{2}$. Try to find the value of $\sqrt{(a+b)(b+c)(c+d)(d+a)}$. (Express the result in terms of $m, n$)
Solution: $\because a c=b d$,
$\therefore \frac{a}{b}=\frac{d}{c}=\frac{a+d}{b+c} \cdot(b+c \neq 0$, otherwise it leads to $m=0$ )
Thus, $\frac{a+b}{b}=\frac{m}{b+c}$,

which means $(a+b)(b+c)=b m$.
Similarly, $(c+d)(d+a)=d m$.
Therefore, the original expression $=\sqrt{b d m^{2}}=\sqrt{n^{2} m^{2}}=m n$.