CN-TST 2016 Problem 11

Does there exist two infinite positive integer sets $S,T$ , such that any positive integer $n$ can be uniquely expressed in the form $$n=s_1t_1+s_2t_2+\ldots+s_kt_k$$ ,where $k$ is a positive integer dependent on $n$ , $s_1<\ldots<s_k$ are elements of $S$ , $t_1,\ldots, t_k$ are elements of $T$ ?