commutative algebra – Contraction of the square of a prime ideal in $mathbb C[x,y,w]/(w^2+x^3+xy^3)$


Consider the $2$-dimensional Noetherian normal domain $R=mathbb C(x,y,w)/(w^2+x^3+xy^3)$ .

Consider the height $1$ prime ideal $P=(w,x)R$ of $R$. Then $P^2R_P=(w^2, x^2, wx)R_P=(xy^3, x^2, wx)R_P=xR_P$ .

How to show that $P^2R_P cap R$ is a principal ideal of $R$ ?