Example 3. How many non-negative integer solutions does the equation $x+3 y+4 z=665$ have?

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
Let $x=12 x_{1}+r_{1}\left(0 \leqslant r_{1} \leqslant 11\right)$,
$$
\begin{array}{l}
y=4 y_{1}+r_{2}\left(0 \leqslant r_{2} \leqslant 3\right), \\
z=3 z_{1}+r_{3}\left(0 \leqslant r_{3} \leqslant 2\right) .
\end{array}
$$

The equation becomes
$$
\begin{array}{ll} 
& 12\left(x_{1}+y_{1}+z_{1}\right)+r_{1}+3 r_{2}+4 r_{3}=665 . \\
& \therefore r_{1}+3 r_{2}+4 r_{3} \equiv 665 \equiv 5(\bmod 12) . \\
& \because 0 \leqslant r_{1}+3 r_{2}+4 r_{3} \leqslant 28, \\
& r_{1}+3 r_{2}+4 r_{3}=5 \\
\text { or } & r_{1}+3 r_{2}+4 r_{3}=17 .
\end{array}
$$

It is easy to find that (1) has 3 solutions, (2) has 8 solutions.
$\therefore$ The original equation has $3 C_{\frac{655 ~ 5}{2}-2}^{12}+8 C_{\frac{665-5}{2}-1+2}^{12}$ $=17108$ non-negative integer solutions.