Decomposition of a $2$-connected graph into three connected subgraphs

Let $G$ be a $2$-connected graph on $n$ vertices such that any its $2$-vertex cutset splits $G$ into at most three parts and $n_1+n_2 +n_3=n$. We prove that there exists a decomposition of the vertex set of $G$ into three disjoint subsets $V_1$, $V_2$, $V_3$, such that $|V_i|=n_i$ and the induced subgraph $G(V_i)$ is connected for each $i$.