A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines \( L_1: 2x + y + 6 = 0 \) and \( L_2: 4x + 2y - p = 0, \, p > 0 \), at the points \( A \) and \( B \), respectively. If \( AB = \frac{9}{\sqrt{2}} \) and the foot of the perpendicular from the point \( A \) on the line \( L_2 \) is \( M \), then \( \frac{AM}{BM} \) is equal to:
