$(15$ points$)$ Consider the CSP where there are four variables $x,y,z,w$ with domain $D=\{1,2,3\}$ and

they have the following constraints.

$x+y2,x+w3$

$y+z4,z+w2$

$y+w5,$

$($a$)(3$ points$)$ Draw the constraint graph for this CSP

$($b$)(12$ points$)$ Before starting our backtracking search for a solution, we run the AC$3$ Algorithm on

this instance, with the initial edge queue below:

$(x,y),(x,w),(y,x),(y,z),(y,w),(z,x),(z,y),(z,w),(w,x),(w,y),(w,z)$

Describe the run of the algorithm for this instance: state what domains are reduced $($write none if

unchanged$)$ and what edges are added to the end of the queue in every iteration. You may write

your answer by filling a table like this one.

What will the algorithm output?