Tournament Matchmaking

You are organizing a recreational rugby tournament. A rugby team has $15$ distinct roles, numbered $1$ to $15$. Each team in the tournament must have exactly $15$ players, each fulfilling one of the roles. Although several groups of friends showed up to play in the tournament, none of the groups are large enough to form a complete team. You would like to create teams by merging some pairs of groups together.

Each group has between $1$ and $14$ players (inclusive) and you know that each player has exactly $2$ potential roles they could play on a team. Determine the maximum number of valid teams you can form. A team is valid if it is made of exactly two groups, it has exactly $15$ players (no more, no fewer), and every role on the team is played by a different player able to play that role. A group cannot be part of more than one team.

The first line contain a single integer $n (1 \le n \le 500)$, the number of groups.

Following this line are $n$ group descriptions. The first line of a group description contains a single integer $k (1 \leq k \leq 14)$, the size of the group. The following $k$ lines each contain two space-separated integers $a$ and $b$ $(1 \leq a < b \leq 15)$, representing a player that can fulfill either role $a$ or role $b$ on team.

Output the maximum number of valid teams that can be created by merging pairs of groups together.