"Go a long way to" + gerund vs infinitive

Which one is correct? If they all are correct, which construction is the most preferable? Why?

1. The fund will go a long way to solving their problem.
2. The fund will go a long way to solve their problem.
3. The fund will go a long way toward(s) solving their problem.
4. The fund will go a long way in solving their problem.

Solution 1:

Number 3 is correct.

You may find Number 1 used on occasion, but it would be considered incorrect by most people.

EDIT:Comments ask about the difference between 'to' and 'towards'. Going 'towards' somewhere means I am going in that direction, but going 'to' somewhere implies I will get there. To go 'a long way towards' means I'm not going to reach there, and therefore 'towards' is right and not 'to'. The same applies metaphorically.

There is a construction "I am going a long way to somewhere", but it means 'taking a long route', and is different from the sense the question is asking about.

2 is wrong for the same reason as 1 (among others) and 4 is wrong because 'in' is the wrong preposition here.

Solution 2:

The best choice is 3. It means that the fund will carry someone a considerable distance toward achieving a goal.

Number 2 is grammatically correct, but the meaning becomes potentially ambiguous because to has many definitions in addition to toward. Some readers or listeners might at least initially interpret to as meaning in order to, suggesting that, in the sentence's meaning, the fund is exercising its own motivation to solve the problem. In reality, people often use such language however, especially in speech, and context usually renders the intent so clear that we often won't notice the potential problem.

Number 1 is probably acceptable to most, and I think this kind of pattern is commonly used. Good style would have us more often use toward/s instead of to, for clarity. Again, a reader or listener would need to decide whether to means toward or for the purpose of.

Number 4 is grammatically correct, but would carry yet another meaning: After beginning to somehow solve their problem, the fund will travel a long distance (somewhere). Compare this to: She will have many adventures in sailing the seven seas.

The question header raises the common problem of selecting between a gerund (verbal noun; -ing form) after the first verb when we have verb + verb. Searching for infinitive or ing will yield many examples, and searching for infinitive or ing preposition will yield many examples of the rule that we use -ing forms, not full infinitives, after prepositions (such as to and toward. That common issue does not exactly apply here. The choice that is often treated in grammars is full infinitive [to + verb] or -ing form. Here, (in 1 vs. 2) you are considering a choice between an -ing form, solving, and the bare infinitive solve.