Theorem $2$ shows that the amortized running time of the Move$-$to$-$Front $($MF$)$ heuristic on linear lists is within a constant $($i$.$e$.,2)$ multiplicative factor of the best off$-$line strategy. Does the same result hold $($possibly with a somewhat larger multiplicative constant factor$)$ for the Move$-$Half$-$Way$-$to$-$Front $($MHWF$)$ strategy? Prove your claim. $($The latter strategy moves the accessed$/$inserted item half$-$way towards the front. That is$,$ if the item was at position i$,$ then it is moved to position $$i$/2,$ without affecting the relative order of the other elements.$)$

$[$Hint: First compare MHWF with MF$,$ then apply transitivity.$]$