6. Eight people participate in a round-robin tournament (i.e., each pair competes in one match). Each match awards 2 points to the winner, 0 points to the loser, and 1 point to each in the case of a draw. After all matches are completed, it is found that:
(1) The eight participants have different scores;
(2) The second-place score is equal to the sum of the scores of the fifth, sixth, seventh, and eighth-place participants;
(3) The first-place participant did not defeat the fourth-place participant.
Based on the above information, the third-place participant scored $\qquad$ points.
$10$