A way to solve nonsmooth stiff ODEs

Question

Let us considered the following ODEs \begin{align*} \dfrac{dX}{dt} = F(X), \tag{1.1} \end{align*} where the unknown $X\in \mathcal{D} \subset \mathbb{R}^l$ and it can be stiff. These ODEs can be solved numerically by classical Newton's method if $F$ is differentiable almost everywhere on the considered domain \begin{align*} \dfrac{X^{n+1}-X^n}{\Delta t} = F\left(X^{n+1}\right). \end{align*} In cases of $F$ is not differentiable everywhere, I have read in a lecture that there is a way to solve numerically by replacing the system (1.1) with the following system \begin{align*} X^{n+1} - X^n - (1-\nu)\Delta t F(X^{n+1}) & = 0 \in \mathbb{R}^l\tag{1.2a}\\ \det\left[I - (1-\nu)\Delta t \nabla F(X^{n+1}) \right] - w &= 0 \in \mathbb{R} \tag{1.2b} \\ \min(\nu,w) &= 0 \in \mathbb{R} \tag{1.2c} \end{align*} where $\nu$ and $w$ are two new unknowns to be sought for simultaneously to $X^{n+1}$. I wonder how they can do like this and what is technique they use?