Question 99
For a positive integer $n$, denote with $b(n)$ the smallest positive integer $k$, such that there exist integers $a_1, a_2, \dots, a_k$, satisfying $n = a_1^{33} + a_2^{33} + \dots + a_k^{33}$. Determine whether the set of positive integers $n$ is finite or infinite, which satisfy:
$$
\text{a) } b(n) = 12; \quad \text{b) } b(n) = 12^{12^{12}}.
$$
$$
\text{a) } b(n) = 12; \quad \text{b) } b(n) = 12^{12^{12}}.
$$