Reversing a Snake on a Complete Graph

A game company decided to build a new version of the classic game Snake, played on a complete graph. A complete graph is a graph in which every two distinct vertices are joined by an edge.

In this game a snake of length $M$ is a directed path $v_1, v_2, \dots, v_M$ through $M$ distinct vertices of the complete graph, where $v_1$ is the head and $v_M$ is the tail; consecutive vertices are joined by an edge.

A single move works as follows. The head $v_1$ moves to some vertex $u$ that is currently not occupied by the snake (empty). The whole body then shifts forward by one, the tail cell $v_M$ becomes empty, and the snake becomes $u, v_1, \dots, v_{M-1}$. Because the graph is complete, $u$ may be any currently empty vertex. (The head may not move onto a vertex occupied by the body or the tail.)

A mathematician proved that if the snake can be reversed in finitely many moves — that is, made to occupy exactly the same edges while the head and the tail swap places — then this is equivalent to being able to reach any arrangement of the snake on the graph. Testing the game therefore reduces to deciding, for a given starting arrangement, whether the snake can be reversed.

Reversing the snake means performing moves until the snake becomes $v_M, v_{M-1}, \dots, v_1$ (the starting path read backwards).

For each scenario, decide whether the snake can be reversed.

The first line contains the number of scenarios $T$ ($1 \le T \le 100$).

Each scenario follows. A scenario consists of the number of vertices $N$ ($3 \le N \le 100$), the snake length $M$ ($2 \le M \le N$), and then $M$ distinct vertex numbers describing the snake from head to tail. Vertices are numbered from $1$ to $N$.

For each scenario, print on its own line YES if the snake can be reversed, or NO otherwise.