$($c$)$ Continuing with the example for part $($b$)$ above, consider the

case where the $A^{**}$ algorithm is to be used to compute a path

from node $1$ to node $3.$ If the heuristic cost to go vector is

represented as shown below, again where the values have been

replaced with letter identifiers, what should the maximum value

be for elements $h_1,h_5,$ and $h_7,$ for the algorithm to be guaranteed

to return an optimal result? Explain your answer.

$heuristi{c}_{cos{t}_t{o}_{g}o}=\left[\begin{array}{c}{h}_1\\ h_2\\ h_3\\ h_4\\ h_5\\ h_6\\ h_7\\ h_8\end{array}\right]$

$($d$)$ Provide a brief explanation of why the algorithm requires these

maximum limits on the heuristic cost to go values.Do not use chatGTP please provide detailed explanaion for each answer, ALSO complete entire cost matrix if possible.