quartic plane curves and their flexes

Question

Suppose $C$ is a smooth, irreducible, quartic plane curve on the complex projective plane and let $P\in C$ be a flex (inflection point) on the curve. Is it true that any plane algebraic curve $C'$ (possibly reducible and singular) of degree $\ge 4$ that intersect $C$ at $P$ will intersect $C$ at another point? Note: you aren't allow to take "multiples of a curve" to get this curve of "degree $\ge 4$"

Answer 1

No. Let, for instance $C$ be $$ C = \{yz^3 + y^4 = x^4\} $$ and $C' = \{yz^3 = x^4\}$. Then their only intersection point is $P = (0,0,1)$, which is a flex point of $C$.