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 \( T_n \), and \( T_n \) 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?