Exact limit of the cube-root analog of the mysterious pattern

Determine the exact value of the limit of the properly convergent sequence a_n defined by a_n = 3^n ∛(3 − f^{(n)}(t_0)), where f(t) = ∛(24 + t) and t_0 is any admissible initial value in the contraction domain (for example, t_0 = 0).

Background

To parallel the original nested-square-root pattern, the authors consider f(t) = ∛(24 + t) with fixed point L = 3 and derivative |f'(L)| = 1/27, and define a_n = 3^n ∛(3 − f^{(n)}(t_0)). They show the sequence converges properly, serving as a natural analog of the mysterious pattern.

Despite establishing proper convergence, the exact value of the limit is not known for this cube-root analog, prompting a concrete unresolved question about its evaluation.