Question $3.$ Backtracking Search.

Consider the graph with $8$ nodes $A_1,A_2,A_3,A_4,H,T,K,F_1,F_2.A_i$ is connected to

$A_{i+1}$ for all $i,$ each $A_i$ is connected to $H,H$ is connected to $T,$ and $T$ is connected to each $F_i,$

and $K$ is connected to $H$ and $F_2.$ Find a $3-$coloring of this graph by hand using the following

strategy: backtracking, the variable order $A_1,H,A_4,F_1,A_2,F_2,A_3,T,K$ and the value order

$R,G,B.$