Example 1. If $m$ satisfies the equation
$$
\begin{array}{l}
\sqrt{3 x+5 y-2-m}+\sqrt{2 x+3 y-m} \\
=\sqrt{x-199+y} \cdot \sqrt{199-x-y},
\end{array}
$$

try to determine the value of $m$.
Analyzing this problem, we have one equation with three unknowns, so we can only solve it cleverly. Observing the characteristics of each algebraic expression in the original equation, we know that
$$
x-199+y \geqslant 0,
$$

and $199-x-y \geqslant 0$.
Thus, $x+y \geqslant 199$,
and $x+y \leqslant 199$.
Therefore, $x+y=199$.
Substituting $x=199-y$ into the original equation, we get
$$
\sqrt{2 y-m+595}+\sqrt{y-m+398}=0 \text {. }
$$

From (2), we know that it can only be $2 y-m+595=0$ and $y-m+398=0$. Eliminating $y$, we get $m=201$.