3. Given that $x, y, z$ are real numbers, and satisfy
$$
x+2 y-5 z=3, x-2 y-z=-5 \text{. }
$$

Then the minimum value of $x^{2}+y^{2}+z^{2}$ is ( ).
(A) $\frac{1}{11}$
(B) 0
(C) 5
(D) $\frac{54}{11}$
Hint: Use the method of elimination.
When $z=\frac{1}{11}$, the minimum value of $x^{2}+y^{2}+z^{2}$ is $\frac{54}{11}$.