Let \( ABC \) be an equilateral triangle with orthocenter at the origin and the side \( BC \) on the line \( x + 2\sqrt{2}\,y = 4 \). If the co-ordinates of the vertex \( A \) are \( (\alpha, \beta) \) then the greatest integer less than or equal to \( \lvert \alpha + \sqrt{2}\beta \rvert \) is:
