Every finitely generated module is sum of local modules

Question

Note: All modules are over a $K$-algebra $A$ with $K$ a field and the underlying ring of $A$ is unital (but not necessarily commutative).

Definition: A module $V$ is local if there is a maximal submodule $U$ of $V$ such that all proper submodules of $V$ are contained in $U$.

Show that every module $V$ of finite length is the sum of all local submodules.

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I'm completely stumped. What I know is that

1. a module of finite length is finitely generated,
2. in a finitely generated module, every proper submodule is contained in some maximal submodule,
3. a lot of facts about socles and radicals (this might help).

Maybe one could try to show that if $L$ is the sum of all local submodules, then $V = \mathrm{rad}(V) + L$. Since the radical is small in the finitely generated $V$, we have $L = V$.