Given that there are $24$ primes between $3$ and $100$, inclusive, determine the number of ordered pairs $(p, a)$ where $p$ is a prime number such that $3 \le p < 100$, and $1 \le a < p$. The condition is that the sum:
\[
a + a^2 + a^3 + \cdots + a^{(p-2)!}
\]
is not divisible by $p$.

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