Question $3[4$ pts$]$: Figure $3$ shows a search tree where A denotes the node corresponding to the initial state, and $E$ is the goal node. In the figure, $h=x$ denotes the heuristic function value and $c=x$ denotes the actual cost between nodes $($i$.$e$.,$ arch cost$).$

a$.$ Please explain why $($or why not$)$ the heuristic function in Figure $3$ is $($or is not$)$ an admissible heuristic $(0\mathrm{.}5pt.$

b$.$ Please explain why $($or why not$)$ the heuristic function in Figure $3$ is $($or is not$)$ a consistent heuristic $(0\mathrm{.}5pt.$

c$.$ Correct heuristic values in Figure $3,$ such that $A^{**}$ search using corrected heuristics can find optimal path from A to E without revisiting nodes. Solutions must show corrected heuristic values for all nodes $(1\mathrm{.}5pt)$

d$.$ Create a search tree using corrected heuristic values to carry out A$*$ search from A to find goal node E $($Node expanded$/$visited does not need to be revisited$).$ Show the search tree $($The tree must show the complete search process with each node's $f(N),g(N),$ and $h(N)$ values$),$ and report the order of the nodes being expanded, and the final discovered path from $A$ to $E(1\mathrm{.}5$ pt$)$