Let the foot of perpendicular from the point $\mathrm{A}(4,3,1)$ on the plane $\mathrm{P}: \mathrm{x}-\mathrm{y}+2 \mathrm{z}+3=0$ be N . If $\mathrm{B}(5, \alpha$, $\beta$ ), $\alpha, \beta \in \mathbb{Z}$ is a point on plane $P$ such that the area of the triangle $A B N$ in $3 \sqrt{2}$, then $\alpha^2+\beta^2+\alpha \beta$ is equal to $\_\_\_\_$ .