Let the foot of perpendicular from the point A (4,3,1) on the plane P: x-y+2 z+3=0 be N. If B(5, α, β), α, β ∈ Z is a point on plane P such that the…

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