A first or the first after mentioning general categories
Solution 1:
There is only one first approach, so the definite article the should be used.
If you said a first approach, I would understand a first (possible) approach, that is, that more than one approach could be considered as claiming the first place in the explanation.
In your edit, you yourself use the:
In total, there are 3 approaches, and the 2nd sentence above is referring to the 1st.