Let \( x = 2t,\, y = \frac{t^2}{3} \) be a conic. Let \( S \) be the focus and \( B \) be the point on the axis of the conic such that \( SA \perp BA \), where \( A \) is any point on the conic. If \( k \) is the ordinate of the centroid of \( \triangle SAB \), then \( \lim\limits_{t \to 1} k \) is equal to:
