Let \( S \) be the set of all \( a \in \mathbb{N} \) such that the area of the triangle formed by the tangent at the point \( P(b,c) \), \( b,c \in \mathbb{N} \), on the parabola \( y^{2}=2ax \) and the lines \( x=b \), \( y=0 \) is \( 16 \, \text{unit}^2 \), then \( \sum_{a \in S} a \) is equal to ________.
