How do you determine the specific compound propositions you need to create from given premises when the question only has imply statements??

Question

Prove validity question:

1. u -> p
2. q -> (u ^ s)
3. (p ^ s) -> r Therefore, q -> r

The strategy used by the answer is creating (u ^ s) -> (p ^ s).

However, I have trouble path finding my way to that using u -> p.

The answer is this:

Note: all numbers here (e.g. (1)), are in the context of this answer section.

1. Create ~u v p v ~s by appending ~u v p with ~s through generalisation
2. Create ~u v s v ~s through the "spawning" of ~u v t
3. Attach (1) and (2) together to produce ~u v p v ~s ^ ~u v s v ~s using conjunction.
4. Distribute (3) into (~u v ~s) v (p ^ s) and so on....

Questions:

1. (1) How do you determine that you have to attach the negation of s in (1)?
2. (2) How do you even know you have to use s in the first place and use the negation of it?
3. (3) How do you determine that you need (2) in that specific format (~u v s v ~s)?
4. (4) How do you determine that you need to use a conjunction to attach (1) and (2)?
5. (5) How do you distribute (3)?

Summary: I am trying to determine what one is seeing to know that (1) and (2) are what you need and (3) is what you need to do because I cannot see anything that tells me I need to do that except that I need to get (u ^ s) -> (p ^ s).

I asked my lecturer, and all it says is "It is not intuitive but it works. Hope this helps.". Not very helpful. I tried inverting from it the strategy, I got lost. I tried to start from what is given in the question and the rules I was given, I don't see how I can determine how I know I need to create (1), (2), and (3). I was hoping people here could explain what I am missing to close this gap.

Thanks.