This is my first submission to a competition! :D

This works by only removing an input once every 210 tapes.
This is accomplished by 3 arms that all have to pivot the input 3 times on the same tape iteration in order for a fourth arm to grab the input and extract it for processing.
The first arm repeats state every 6 tapes.
The second arm repeats state every 5 tapes.
the third arm repeats state every 7 tapes.
Since each of these numbers have separate prime factors (2x3, 5, 7), the least common multiple of these numbers is 210.

Using arms with AND-style logic was the only approach I felt I was capable of pulling off. I considered the possibility of inserting a 6x multiplier with the Berlo wheel spinning so that output would only match every 1/6 times, but I realized that would require me to change the input grabber to not grab if the output had not yet been processed, and I'm just not ready for the level of wanding shenanigans that would take, if it is possible in this small space.

I think this approach could be improved with better space utilization and another arm repeating either every 11 cycles or taking the 6 cycle arm and replacing it with a 4 cycle arm and a 3 cycle arm (I expect the primes would stack if they're on the same arm). The difficulty with the escalating size in primes is that they require tracks whose size equal their repeating period. Trying to put multiple primes on one arm means multiplying the primes across each other, which takes up exorbitantly more space.