resolution of curve singularities

Let $\pi:X\longrightarrow C$ be the minimal good resolution of the curve singularity (C,o) with exceptional set $E$, where $C:=\{x_1x_2(x_1^{a_1}+x_2^{a_2})=0\}\subset \mathbb C^2$. Let $\bar C_i$ be the strict transform of $\{x_i=0\}$ for $i=1,2$. Then $\bar C_1$ and $\bar C_2$ intersect distinct ends of $E$ if $E$ is not irreducible??

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