Let . Find the last three digits of the number of distinct positive reals satisfying .
Details and Assumptions
denotes the function applied times successively on , i.e. .
Any non-real / repeating solutions of the equation should be ignored. For example, the roots of the equation (counted with multiplicity) are , but its number of distinct real solutions is .
This problem appeared in the Proofathon Algebra contest, and was posed by me.