In a club, some pairs of members are friends. Given an integer $k \geqslant 3$, we say a club is $k$-friendly if, in any group of $k$ members, they can be seated at a round table such that each pair of neighbors are friends. Prove that if a club is $6$-friendly, then it is also $7$-friendly. Is it true that if a club is $9$-friendly, then it is also $10$-friendly?