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