Suppose you have the following priority min-heap: [4, 9, 7, 10, 12, 8, 11, 15, 32]

(a). Decrease the key of 32 to 2 and recompile the min-heap in O(lg n) time. Show your work.

(b). After applying part (a), add a new element 3 to the heap in O(lg n) time. Show your work.

(c). After parts (a) and (b), remove the minimum element and recompile the min-heap in O(lg n) time. Show your work.