To break down a problem

There exists a problem or goal which cannot be achieved in one action, therefore one breaks the problem into smaller problems and solves each to ultimately reach the goal.

In one word, what verb is one doing to the problem?

It is called problem decomposition or decomposing a problem. According to State Uni. Of New York

Problem decomposition is the problem solving strategy of breaking a problem up into a set of subproblems, solving each of the subproblems, and then composing a solution to the original problem from the solutions to the subproblems.

Definition of decompose according to Merriam Webster is:

to separate into constituent parts or elements or into simpler compounds

Well, you're breaking down the problem, but if you need one word, I'd go with dissect:

1: to separate into pieces
: expose the several parts of (something, such as an animal) for scientific examination
// dissect an earthworm
// dissecting flowers
2: to analyze and interpret minutely // dissect a problem

(source: Merriam-Webster)

You see how the first (literal, and probably older) definition leads to the second (figurative) definition.