In a tournament, there are \( n \) teams \( T_1, T_2, \ldots, T_n \), with \( n>5 \). Each team consists of \( k \) players, \( k>3 \). The following pairs of teams have one player in common:
\( T_1 \) and \( T_2 \),
\( T_2 \) and \( T_3 \),
...
\( T_{n-1} \) and \( T_n \),
\( T_n \) and \( T_1 \).
No other pair of teams has any player in common.
How many players are participating in the tournament, considering all the \( n \) teams together?