162 On the Cartesian plane, a point whose both coordinates are integers is called an integer point. For some integer point $P$ and some positive number $d$, there are exactly $k(>0)$ different integer points, each at a distance $d$ from $P$. Denote the range of $k$ values as $\left\{k_{1}, k_{2}, \cdots\right\}$, where $0<k_{1}<k_{2}<\cdots$, then $k_{2}=$ $\qquad$
162 8. Translate the coordinate system in parallel so that $P$ becomes the origin, then these $k$ integer points remain integer points in the new coordinate system.
Let integer point $A_{1}(x, y)$ be one of the $k$ integer points. Note that $A_{2}(-x,-y), A_{3}(-x, y)$, $A_{4}(x,-y), A_{5}(y, x), A_{6}(-y, x), A_{7}(-y,-x), A_{8}(y,-x)$ are all integer points (some may coincide), and their distances to point $P$ are all $d=\sqrt{x^{2}+y^{2}}$. Therefore, $K_{1}=4, K_{2}=8$.