Consider a circle \( (x-\alpha)^{2} + (y-\beta)^{2} = 50 \), where \( \alpha, \beta > 0 \). If the circle touches the line \( y + x = 0 \) at the point \( P \), whose distance from the origin is \( 4\sqrt{2} \), then \( (\alpha + \beta)^{2} \) is equal to __________.
