Consider a string of $$n$$ $$1$$'s. We wish to place $$+$$ signs in between so that the sum is $$1000$$. For instance, if $$n=190$$, one may put $$+$$ signs so as to get $$11$$ ninety times and $$1$$ ten times, and get the sum $$1000$$. If $$a$$ is the number of positive integers $$n$$ for which it is possible to place $$+$$ signs so as to get the sum $$1000$$, then find the sum of the digits of $$a$$.