Each team plays $2$ games.  $1$ points: a lost and a tie.  $2$ points: two ties.  $3$ points: a lost and a win.  $4$ points: a win and a tie.  $6$ points: two wins. 
Three football teams participate in a sport tournament. Each team plays the other two teams exactly once. In each game, the winner gets three points and the loser doesn\textquotesingle t get any points. If the game ends in a tie, each team gets $1$ point. At the end of tournament, Which number of points is it impossible for any team to have? (2022 Math Kangaroo Problem, Level 3-4, Question \#18) 

$$1$$ 
$$2$$ 
$$4$$ 
$$5$$ 
$$6$$ 