Consider the following $2$SAT problem with clauses $\{x_j,not{x}_{j+1}\}$ for $j=1,2,$dots,$100,$ along with the additional clause

$\{x_{100},x_1\}$

Which of the following statements are true? $($more than one might be valid$)$

The implication graph will contain a "bad loop" indicating that the original $2$SAT problem is unsatisfiable.

The additional clause $\{x_{100},x_1\}$ will introduce a direct edge from $not{x}_{100}$ to $x_1$ and from $xno{t}_1$ to $x_{100}$ in the graph

The implication graph contains $200$ directed edges.

There will be a direct edge from $x_{j+1}$ to $x_j$ for each $j$ from $1$ to $100.$

The implication graph contains $100$ edges, as there are $100$ clauses.

There will be a direct edge from $not{x}_j$ to $not{x}_{j+1}$ for each $j$ from $1$ to $100.$

There is a path from $not{x}_1$ to $not{x}_{100}$ and then to $x_1$ in the implication graph. However, there is no path from $x_1$ back

to $not{x}_1.$ Thus, the implication graph does not contain a bad loop, and the clauses are satisfiable.

The implication graph will have $200$ nodes.