Birthday Gifts

It’s Chef's birthday. He and his twin have received $N$ gifts in total. The $i$-th gift has a price of $A_i$. Each twin wants to keep the most expensive gifts for himself. The twins take $K$ turns alternately (each has $K$ turns, for $2\cdot K$ turns in total). It is given that $2\cdot K+1 \leq N$. In a turn, a person may choose one gift. The only exception is the last turn of the twin who mov