Making a CFG for $a^{i}b^{j}c^{k}$ such that $i+j=k+3, i,j\geq1$

Question

I have the language $$ L=\{a^{i}b^{j}c^{k} \mid i+j= k+3, i,j\geq1\} $$however I am struggling to convert it to a CFG. I ended up with this grammar: \begin{align} S &\rightarrow aSc \mid aBAc \mid B\\ A &\rightarrow bAc \mid bc \\ B &\rightarrow aaa \mid aab\mid abb\mid bbb \\ \end{align}

Is it correct?