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Here's an example with $|B_1|=1, |B_2|=2$: Let $G=\mathbb Q*\mathbb Q$ (free product). Write non-identity elements of $G$ multiplicatively as reduced words with $X^a, Y^b: a,b \in \mathbb Q - \{0\}$ (with symbols $X,Y$). Let $A=$ (non-identity) reduced words with all non-integers (i.e. reduced words with $X^{\mathbb Q - \mathbb Z}, Y^{\mathbb Q - \mathbb Z}), B_1=\{1\}, B_2=\{X,Y\}$. Note we can't have $A \cdot B_1=A \cdot B_2=G$ with $|B_1|=1, |B_2|>1$.