limit of $x_{n+1}=x_{n}^{n+1}+x_{n}+1$

Question

Let $(x_{n})_{n\geq1 }$ be a sequence of real numbers such that $x_{1}> 0,x_{n+1}=x_{n}^{n+1}+x_{n}+1, \forall n\geq 1$. Show that $\exists a\epsilon \mathbb{R}, a\neq 0$ for which $\lim_{n\rightarrow \infty }\frac{x_{n}}{a^{n!}}=1$. I showed that the sequence diverges to infinity, but I don't know what to do further. I tried to apply the Stolz Cesaro theorem, but there has to be something more I do not notice here.