Partitioning the positive integers into finite sets with sums in geometric progression

Here is a quite interesting problem I've come up with:

Let $\mathbb{N}^+ = \{1, 2, 3, \dots\}$.

Does there exist a sequence of sets $A_1, A_2, \dots$ such that:

$1$. $A_k \subset \mathbb{N}^+$ is nonempty and finite.

$2$. $A_i \cap A_j = \varnothing$ for $i \neq j$.

$3$. $\bigcup_{k=1}^{\infty} A_k = \mathbb{N}^+$.

$4$. The sums of the elements in each group form a geometric progression, i.e. $S_k = \sum_{x \in A_k} x = a r^{k-1}$ for some $a, r > 0$.

I'm curious whether such a partition exists. I would appreciate a proof or any ideas.