Find
$$\large \lim_{n \to \infty}\sqrt[3]{a_0+\sqrt[3]{a_1+\cdots+\sqrt[3]{a_n}}},$$ where $\large a_n=6(3^{3^n}+1)^2.$
According to Herschfeld's Convergence Theorem, the limit do exists , since $\large\dfrac{\ln a_n}{3^n}$ is bounded. But how to compute it?