8 Positive integers $x, y, z$ satisfy $\left\{\begin{array}{l}7 x^{2}-3 y^{2}+4 z^{2}=8, \\ 16 x^{2}-7 y^{2}+9 z^{2}=-3 .\end{array}\right.$ Find the value of $x^{2}+y^{2}+z^{2}$.
8. Let $x, y, z$ satisfy
$$\left\{\begin{array}{l}
7 x^{2}-3 y^{2}+4 z^{2}=8 \\
16 x^{2}-7 y^{2}+9 z^{2}=-3
\end{array}\right.$$

Multiplying (1) by 7 and (2) by 3, we get

Substituting back into (1) yields
$$\begin{array}{c}
x^{2}+z^{2}=65 \\
3 x^{2}-3 y^{2}=8-260
\end{array}$$

Thus,
$$\begin{aligned}
y^{2}-x^{2} & =84 \\
(y-x)(y+x) & =2^{2} \times 3 \times 7
\end{aligned}$$

Since $y-x$ and $y+x$ have the same parity, we have
$$(y-x, y+x)=(2,42),(6,14)$$

Solving these, we get
$$(x, y)=(20,22),(4,10)$$

However,
$$x^{2}+z^{2}=65$$

Thus, it can only be
$$(x, y)=(4,10)$$

In this case,
$$z=7$$

Therefore,
$$x^{2}+y^{2}+z^{2}=165$$