After operations $$$1,\ldots,i$$$, view the current permutation as an interleaving of two subsequences. The processed values $$$1,\ldots,i$$$ must already appear in their relative order in $$$P$$$, because they will never move again. The untouched values $$$i+1,\ldots,N$$$ still appear in increasing relative order. At operation $$$i+1$$$, track how that value moves from the untouched subsequence to the processed one.

Start from $$$P$$$ and undo the moves in the order $$$N,N-1,\ldots,1$$$. To undo the move of $$$x$$$, remove $$$x$$$ from the processed subsequence and place it before all values of the untouched subsequence $$$x+1,\ldots,N$$$, making $$$x$$$ its first element. How does the position of this first untouched element change as you continue undoing the moves?