Effective lower bound for the greatest prime factor

Derive an effective lower bound for the greatest prime factor of the integers x_m defined in Theorem 1, where x_m is the increasing sequence of integers with base-b representations having factor complexity at most Cℓ and satisfying the stated nondivisibility condition, and where the prime divisors of x_m are not eventually confined to a fixed finite set.

Background

Theorem 1 proves, by an application of the p-adic Schmidt Subspace Theorem, that for any fixed finite set S of primes, all sufficiently large integers x_m have a prime divisor outside S. Equivalently, the greatest prime factor of x_m tends to infinity. However, the Schmidt Subspace Theorem is ineffective, so the proof does not provide an explicit threshold or quantitative rate for this growth. The unresolved problem is to obtain an effective lower bound for the greatest prime factor of x_m.