Compute limit of the sequence $x_n$ given by $x_{n+2}=-\frac{1}{2}(x_{n+1}-x_n^2)^2+x_n^4$

Question

Let $(x_n)$ be a real sequence such that $x_0=a\in\mathbb{R},x_1=b\in\mathbb{R},x_{n+2}=-\dfrac{1}{2}\left(x_{n+1}-x_{n}^2\right)^2+x_{n}^4\;\forall n\in\mathbb{N} $ and $|x_n|\leq \dfrac{3}{4},\forall n\in\mathbb{N}$. The sequence $x_n$ is convergence or not and compute $\lim x_n$ (if exists)?

I think we could use $\inf$ and $\sup$. But I am not sure