Start by defining the structure of a node in the linked list that will represent each heap, including the necessary fields to maintain heap properties (e.g., key, child pointers, sibling pointers, etc.).

Implement the `merge` operation by comparing the roots of the two heaps and recursively merging the subtrees, ensuring the heap property (min-heap or max-heap) is preserved throughout the process.

Design the `insert` operation by creating a new heap with a single node and merging it with the existing heap, then optimize the `extract-min` (or `extract-max`) operation by restructuring the heap to maintain efficiency after removing the root.