Proving this sequence is eventually periodic with a period a power of 2

[Sparkson]

This problem was sent to me by my friend.

Let there be a sequence of positive integers a₁, a₂, ....For all a_i, multiply a_i in base 10 by 5²⁰²⁰, replace each digit with its remainder when divided by 2, and read the result as a binary number. Call this new number a_i+1. If a₁ is any positive integer, show that a_k=a_{k+2²⁰²⁰} for large enough k.

I managed to show that any number greater than 2²⁰²¹ will eventually reduce to a number less than 2²⁰²¹, so I can just consider numbers less than 2^2021. I thought of using pigeonhole principle, but that didn't help in showing the period would be a power of 2. I experimented with numbers less than 2020 (like 2, 3, 4), but I couldn't find anything useful.

Thanks in advance.