Mathematics Question from JEE Main 2023 · 31 January, Shift 2

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 \quad \text{and the lines} \quad x = b, , y = 0 \quad \text{is} , 16 , \text{unit}^2, \text{ then} \quad \sum_{a \in S} a , \text{is equal to:}

$$$$