資料結構與演算法

(10%) Consider a binary min-heap represented by the array $[1, 3, 2, 10, 7, 6, 9]$, where the element 1 is at the root. Suppose you need to insert a new element, 4, into this min-heap. Describe the complete insertion process step-by-step. Your answer should include:

• Indicating where the new element is initially placed in the array.
• Explaining the heapify-up (or bubble-up) process in detail, showing each intermediate state of the heap after any necessary swaps.
• Presenting the final min-heap array once the insertion is complete.

Make sure to clearly justify the steps taken and the reasoning behind each comparison and swap, based on the fundamental properties of a min-heap.