Akuya has finally obtained Shinki Hikari's iPhone. Now, he needs to crack the password to gather evidence!
Blindly attempting to crack the password won't work, so Akuya found a note inside Shinki Hikari's phone case. On it was a puzzle! He believes the solution to this puzzle is the phone's password. The puzzle is as follows:
Given a positive integer nnn, the double factorial of nnn is the product of all positive integers with the same odd/even parity as nnn and not exceeding nnn. It is denoted as n!!n!!n!!, for example, 5!!=1×3×5,6!!=2×4×65!! = 1 \times 3 \times 5, 6!! = 2 \times 4 \times 65!!=1×3×5,6!!=2×4×6.
Find the number of trailing zeros in the decimal representation of the product 1!!×2!!×3!!×⋯×n!!1!! \times 2!! \times 3!! \times \dots \times n!!1!!×2!!×3!!×⋯×n!!.
Akuya is just one step away from achieving his revenge. Can you help him?