Ball growth function of a quotient

Let $G$ be a finitely generated group with finite generating set $A$. Define this distance on $G$: $d_{G,A}(g,h)=\min\{n:gh^{-1}=a_1^{\varepsilon_1}\cdots a_n^{\varepsilon_n},a_i\in A, \varepsilon_1=\pm1\}$. Define $B_{G,A}(n)$ the ball of radius $n$ for this distance, and $\beta_{G,A}(n)=|B_{G,A}(n)|$. Let $N\triangleleft G$. I want to prove that there exists a $\lambda$ such that $\beta_{G/N,B}(n)\leq\lambda\beta_{G,A}(\lambda n+\lambda)+\lambda$ where $B$ is a finite generating set for $G/N$. Could you help me solving this problem?

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If $\overline{A}$ denotes the generating set $\{ aN \mid a \in A \}$ of $G/N$, then we have $\beta_{G/N,\overline{A}}(n) \le \beta_{G,A}(n)$.

This reduces the problem to proving that, for two generating sets $A$ and $B$ of the same group $G$, we have $\beta_{G,B}(n) \le \lambda \beta_{G,A}(\lambda n + \lambda) + \lambda$ for some $\lambda$.

To show this, express the elements of $B$ as words in $A$, and let $\lambda$ be the maximum length of these words. Then $\beta_{G,B}(n) \le \beta_{G,A}(\lambda n)$.

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