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Here's an example with $|B_1|=1, |B_2|=2$: Let $G=\mathbb Z*\mathbb Z$ (free product). Write non-identity elements of $G$ multiplicatively as reduced words with $X^{\mathbb Z-\{0\}}, Y^{\mathbb Z-\{0\}} $(with symbols $X,Y$). Let $A_0=X^{\mathbb Z - 2\mathbb Z} \cup Y^{\mathbb Z - 2\mathbb Z}, A_1 = \{$reduced words with $X^{2\mathbb N},Y^{2\mathbb N}\}, A=A_0A_1, B_1=\{1\}, B_2=\{X^2,Y^2\}$. Note we can't have $A \cdot B_1=A \cdot B_2=G$ with $|B_1|=1, |B_2|>1$.

Here's an example with $|B_1|=1, |B_2|=2$: Let $G=\mathbb Z*\mathbb Z$ (free product). Write non-identity elements of $G$ multiplicatively as reduced words with $X^{\mathbb Z-\{0\}}, Y^{\mathbb Z-\{0\}} $(with symbols $X,Y$). Let $A_0=X^{\mathbb Z - 2\mathbb Z}, A_1 = \{$reduced words with $X^{2\mathbb N},Y^{2\mathbb N}\}, A=A_0A_1, B_1=\{1\}, B_2=\{X^2,Y^2\}$. Note we can't have $A \cdot B_1=A \cdot B_2=G$ with $|B_1|=1, |B_2|>1$.

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^{\mathbb Q-\{0\}}, Y^{\mathbb Q-\{0\}} $(with symbols $X,Y$). Let $A_0=\{$ (non-identity) reduced words with all non-integers i.e. reduced words with $X^{\mathbb Q - \mathbb Z}, Y^{\mathbb Q - \mathbb Z}\}, A_1 = \{$reduced words with $X^{\mathbb N},Y^{\mathbb N}\}, A=A_0A_1, B_1=\{1\}, B_2=\{X,Y\}$. Note we can't have $A \cdot B_1=A \cdot B_2=G$ with $|B_1|=1, |B_2|>1$.

Here's an example with $|B_1|=1, |B_2|=2$: Let $G=\mathbb Z*\mathbb Z$ (free product). Write non-identity elements of $G$ multiplicatively as reduced words with $X^{\mathbb Z-\{0\}}, Y^{\mathbb Z-\{0\}} $(with symbols $X,Y$). Let $A_0=X^{\mathbb Z - 2\mathbb Z} \cup Y^{\mathbb Z - 2\mathbb Z}, A_1 = \{$reduced words with $X^{2\mathbb N},Y^{2\mathbb N}\}, A=A_0A_1, B_1=\{1\}, B_2=\{X^2,Y^2\}$. Note we can't have $A \cdot B_1=A \cdot B_2=G$ with $|B_1|=1, |B_2|>1$.

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 $X^{\mathbb Q-\{0\}}, Y^{\mathbb Q-\{0\}}$ (with symbols $X,Y$). Let $A_0=$ (non-identity) reduced words with all non-integers (i.e. reduced words with $X^{\mathbb Q - \mathbb Z}, Y^{\mathbb Q - \mathbb Z}), A=\cup A_0X^n \cup A_0 Y^n : n \ge 0, B_1=\{1\}, B_2=\{X,Y\}$. Note we can't have $A \cdot B_1=A \cdot B_2=G$ with $|B_1|=1, |B_2|>1$.

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^{\mathbb Q-\{0\}}, Y^{\mathbb Q-\{0\}} $(with symbols $X,Y$). Let $A_0=\{$ (non-identity) reduced words with all non-integers i.e. reduced words with $X^{\mathbb Q - \mathbb Z}, Y^{\mathbb Q - \mathbb Z}\}, A_1 = \{$reduced words with $X^{\mathbb N},Y^{\mathbb N}\}, A=A_0A_1, B_1=\{1\}, B_2=\{X,Y\}$. Note we can't have $A \cdot B_1=A \cdot B_2=G$ with $|B_1|=1, |B_2|>1$.