IMO 2020, Problem 2

The following is a question from IMO 2020:

The first time I tried to solve the problem, I thought I had a solution, but it turned out to be wrong. I wrongly assumed that $a^ab^bc^cd^d$ would be maximized when $a=b=c=d$, which is commonly true in Olympiad problems, but that needn’t be the case.

I then looked at solutions available online, and realized that I just needed to show that $a^ab^bc^cd^d\leq a^2+b^2+c^2+d^2$. After doing that, I homogenized both sides, and tried to prove the statement. I was finally successful. I am recording my solution, as it is slightly different from the ones available online.