Limit of a product

Let $(a_n)$ be a sequence defined by $a_1 = 2$ and $a_n = 2a_{n-1}^2 - 1$ for all $n \in \mathbb{N}$, $n \ge 2$. Compute $$\lim_{n \to +\infty} \frac{2^{n-1}(a_{n-1}+1)(a_{n-2}+1)\cdots(a_1+1)}{a_n}.$$