Probability distribution $P$ over binary variables $x_1,$dots,$x_{6}in\{0,1\}$ has the following form:

$P(x_1,$dots,$x_6)=\frac{1}{Z}(x_1,x_2)(x_2,x_3)(x_3,x_4)(x_4,x_5)(x_5,x_6)(x_6,x_1)$

where for $a,$bin$\{0,1\},$

$(a,b)=\{\begin{array}{ccc}1& if& a=b\\ 0\mathrm{.}5& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

$($a$)$ Draw the minimal undirected graph $G$ over which $P$ factors. $($This is the undirected graph with

the fewest possible edges that contains cliques corresponding to each of the terms in $P.)$

$($b$)$ For what configuration$($s$)$ xin$\{0,1{\}}^6$ is $P(x)$ maximized?

$($c$)$ For what configuration$($s$)$ xin$\{0,1{\}}^6$ is $P(x)$ minimized?