I have no idea why neither formulation sounds correct to my ears. What is the problem with ‘completed’. What is the best, alternative and an effective plan?

I want to expand my plan to a Solution to the problem.

(The plan was insufficient, perhaps because it was too’small’, and you have enlarged it enough so that it is now a solution to the problem.) I

developed my plan into a solution.

What would you do if your system had changed?

Is the plan solution a solution/solution to any problem?

As you can see and as you said, when an object (the plan) is transformed by some process (developing, modifying, and so on), the correct particle is into. (to

be used for visualization purposes only)