How many planes are totally tangent to a curve of $\mathbb{P}^3$ which is intersection of a generic quadric and a generic cubic?

Or equivalently, considering the cubic as a blow up of $\mathbb{P}^2$ in $P_1,...,P_6$, how many cubic curves passing through $P_1,...,P_6$ in $\mathbb{P}^2$ are totally tangent to a fixed generic sextic curve with simple nodes at $P_1,...,P_6$?