For every $n \in \mathbb{N}$, let $d(n)$ denote the sum of digits of $n$. It is easy to see that the sequence $d(n), d(d(n)), d(d(d(n))), \ldots$ will eventually become a constant integer between $1$ and $9$ (both inclusive). This number is called the digital root of $n$, denoted by $b(n)$. 

Determine for how many natural numbers $k < 1000$, the limit $\lim_{n \to \infty} b(k^n)$ exists.

Let's think step by step and output the final answer within \boxed{}.
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