I have recently learned the proof of:

$(a\_1 \ a\_2 \ \ldots \ a\_n) = (a\_1 \ a\_k)(a\_1 \ a\_{k-1}) \ldots (a\_1 \ a\_2) = (a\_k \ a\_{k-1}) (a\_k \ a\_{k-2}) \ldots (ak \ a\_1) = (a\_1 \ a\_2) (a\_2 \ a\_3 ) \ldots (a\_{k-1} \ a\_k)$

Now I can prove the above 3 equalities through induction but unfortunately induction, unlike a direct proof, doesn't quite tell me what is really going on. I'm struggling to understand this intuitively. I don't want to just memorize it.

Can anyone walk me through this with an intuitive explanation. Thank you.