There's a cell in (e',1) where the x value is a multiple of one of the q primes and where gcd(a'[t],c) 1. If we knew which q prime it was and what to multiply it by, we'd solve the thing.
I should be more specific. It would appear that the first time a appears as a factor of a'[t], x is a multiple of one of the q primes. And there are infinite cells where a turns up as a factor of a'[t], but this specific cell is the first time that occurs. It doesn't seem obvious at first glance how to predict which of the q primes x is a multiple of and what it's multiplied by.