Avoiding equality of partial sums of two different aperiodic sequences

Question

Consider two distinct sequences of positive integers, $a_{n}|_{n=1}^{\infty}$, and $b_{n}|_{n=1}^{\infty}$ such that for either sequence no period exists. The elements of both sequences are drawn from the same finite list. Consider the partial sums $S_{k}=\sum\limits_{n=1}^{k} a_n$ and $T_{k}=\sum\limits_{n=1}^{k} b_n$.

Does there then have to exist some positive integers $l,m,n$ such that $l=S_{n}=T_{m}$?

In order to avoid such an equality, i.e. for any positive integers $m,n$ to not have any integer $l$ such that $l=S_{n}=T_{m}$, do we have to conclude that the sequences are periodic and should we be forced to conclude that the periods are same for both sequences?

This is a weaker form of a question that came up in a problem on tilings.

Answer 1

Let one sequence be randomly sampled from $\{2,4\}$, and the other obtained similarly after setting the first element to 1. Then these sequences are not periodic and have pairwise distinct partial sums -- for the first sequence they are even, while for the second they are odd.