A straight pillar $PQ$ stands at a point $P$, the points $A$ and $B$ are situated due south and east of $P$ respectively. $M$ is the mid-point of $AB$. $PAM$ is an equilateral triangle and $N$ is the foot of the perpendicular from $P$ on $AB$. Suppose $AN = 20 \text{ meters}$ and the angle of elevation of the top of the pillar at $N$ is $\tan^{-1} (2)$. Find the sum of the angles of elevation of its top at $A$ and $B$.