Example 1 If the real numbers $x, y, m$ satisfy the relation
$$
\begin{array}{l}
\sqrt{3 x+5 y-2-m}+\sqrt{2 x+3 y-m} \\
=\sqrt{x-199+y} \cdot \sqrt{199-x-y},
\end{array}
$$

try to determine the value of $m$.
(1994, Beijing Junior High School Mathematics Competition)
Analysis: An equation with one unknown cannot be solved by brute force, but only by intelligent methods. Observing the characteristics of the equation, all are square roots. By definition, the right side of the equation has
$$
\left\{\begin{array}{l}
x-199+y \geqslant 0, \\
199-x-y \geqslant 0,
\end{array}\right.
$$

which means
$$
\left\{\begin{array}{l}
x+y \geqslant 199, \\
x+y \leqslant 199 .
\end{array}\right.
$$
$$
\therefore x+y=199 \text {. }
$$

At this point, the equation simplifies to
$$
\sqrt{595-m+2 y}+\sqrt{398-m+y}=0 .
$$

Observing this equation again, the left side is the sum of two non-negative numbers, so we have
$$
\left\{\begin{array}{l}
595-m+2 y=0, \\
398-m+y=0 .
\end{array}\right.
$$

Eliminating $y$, we get $m=201$. Then we find $x=2$,
$$
y=197 . \quad \text {. }
$$