$.$ A heap is given below. Answer Q$5\mathrm{.}1$ Q$5\mathrm{.}2$ based on this tree.

$(5\mathrm{.}1)(7$ points$)$ After a new node is inserted, the heap must be modified. Assume that a new node whose value is $97$ has been added to the heap above and the corresponding modifications to the heap also have been done. $(1)$ Write down the algorithm used to insert the node and $(2)$ add the value of each node to the graph representing the heap below.