Let L be a doubly linked list of length n stored in arrays key, prev, and next of length m. Suppose that these arrays are managed by ALLOCATE-OBJECT and FREE-OBJECT procedures that keep a doubly linked free list F. Suppose further that of the m items, exactly n are on list L and m - n are on the free list. Write a procedure COMPACTIFY-LIST (L,F) that, given the list L and the free list F, moves the items in L so that they occupy array positions 1, 2, . . . ,n and adjusts the free list F so that it remains correct, occupying array positions n + 1, n + 2, . . . ,m. The running time of your procedure should be Θ(n), and it should use only a constant amount of extra space. Argue that your procedure is correct.