Let $X$ and $Y$ be indecomposable modules over a finite dimensional algebra and let $f \colon X \to Y$ be a non-zero morphism which is neither a monomorphism nor an epimorphism.

Suppose that it is possible to find a module $Z$ and morphisms $f'' \colon X \to Z$ and $g'' \colon Z \to Y$ such that $$ 0 \longrightarrow X \xrightarrow{(f' f'')} Im f \oplus Z \xrightarrow{(g' g'')} Y \longrightarrow 0 $$ is an exact sequence, where $f'$ is the corestriction of $f$ to its image and $g'$ is the inclusion.

How can I show that this sequence does not split?

Thanks!