How many positive integers $N$ less than $10^{1000}$ are such that $N$ has $x$ digits when written in base ten and $\frac{1}{N}$ has $x$ digits after the decimal point when written in base ten? For example, 20 has two digits and $\frac{1}{20} = 0.05$ has two digits after the decimal point, so $20$ is a valid $N$.

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