Motivation for proof of $(((\phi \to \psi) \to \phi) \to \phi)$ I'm trying to solve this exercise.

If $\phi, \psi$ are propositional formulas, prove $\vdash (((\phi \to \psi) \to \phi) \to \phi)$.

I was wondering if there was a "simple" proof to this (I'm aware of the proof Using the Fitch system, how do I prove $((p \implies q) \implies p)\implies p$?), and most importantly, how to motivate a proof to this exercise. (For me, the motivation is the most important bit! I already have a proof)
I asked such a question before, with a much simpler exercise, and the answer was "try it out in words first, and then write it symbolically!" which worked very well until coming across this example.  Currently I have a proof to this, which I brute forced; to me it feels extremely mechanical and non-replicable but for luck. I'm not sure if this is relevant to the question, but I'll attach it as a picture.

Edit: I actually don't know what the expression "proof system" means. Sorry; I was only told that this was "natural deduction", and the rules for $\land, \lnot, \rightarrow , \leftrightarrow, \perp$, etc were given.

Given the last instance of the material conditional then the last rule to use must be an instance of the deduction theorem (or ($\to I$)), so for the solution a good strategy is to assume the antecedent: $(\phi \to \psi)\to \phi$, turn it into its equivalent disjunctive version: $\lnot(\phi \to \psi) \lor \phi$; then assume the two disjuncts to derive the formula $\phi$ on both sides and dischard the two assumptions $(\lor E)$. then you can make use of $(\to I)$, dischard the original assumption and derive the desired sequent.
