A tree in a Fibonacci heap is $$path$-$like$$ if all of its nodes lie on a single, downward path from the root. Similarly, a Fibonacci heap is $$path$-$like$$ if and only if it contains exactly one tree and that tree is considered $$path$-$like.$$ For this question, you can assume that the keys associated with the Fibonacci heap are arbitrary integers. $***$BE SURE TO EXPLAIN THE STATE OF THE HEAP AND ITS TREES AFTER EACH STEP$***($drawing is encouraged$)$

$($PART A$)$ Exhibit a sequence of four Fibonacci heap operations that transforms the empty Fibonacci heap into a $2-$node path$-$like Fibonacci heap. $($HINT: you will need to add two items, extract min, and perform one other operation to induce correct pairing to form a path$-$like structure$)$

$($PART B$)$ Let H be an i$-$node path$-$like Fibonacci heap, where i $>=2.$ Exhibit a sequence of six Fibonacci heap operations that transforms H into an $($i $+1)-$node path$-$like Fibonacci heap. $($HINT: you will use inserts, extract mins, and decrease key operations$)$