$[2$ pts$]$ Figure $1$ 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$.$

Find all inconsistent heuristic$($s$)$ in Figure $1[0\mathrm{.}25$ pt$]$

b$.$

Create a search tree using the heuristic values given in Figure $1$ to carry out A$*$ search from A to find goal node E $($Node expanded$/$visited does not need to be revisited$).$ The tree must show the complete search process with each node$$s f$($N$)$ values. Report the order of the nodes being expanded, and the final discovered path from A to E $[0\mathrm{.}5$ pt$]$

c$.$

Is the path discovered from the above process optimal or not? Explain why $[0\mathrm{.}25$ pt$]$

d$.$

Correct heuristic values in Figure $1,$ such that the A$*$ using the corrected heuristics can find optimal path from A to E$.$ Solutions must show corrected heuristic values for all nodes $[0\mathrm{.}5]$

e$.$

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$).$ The tree must show the complete search process with each node$$s f$($N$)$ values. Report the order of the nodes being expanded, and the final discovered path from A to E $[0\mathrm{.}5$ pt$]$

Figure $1$: A search tree and husirtic function values.

Figure $1$: A search tree and husirtic function values