GAIC Math 2024 Problem 23

Consider the sequence $\{S_n\}$ constructed as follows: $S_1=\{1,1\}$, $S_2=\{1,2,1\}$, $S_3=\{1,3,2,3,1\}$, and in general, if $S_k=\{a_1,a_2,\cdots,a_n\}$, then $S_{k+1}=\{a_1,a_1+a_2,a_2,a_2+a_3,\cdots,a_{n-1}+a_n,a_n\}$. What is the number of terms equal to $1988$ in $S_{1988}$?