10. (20 points) In an annual super football round-robin tournament, 2013 teams each play one match against every other team. Each match awards 3 points to the winner, 0 points to the loser, and 1 point to each team in the event of a draw. After the tournament, Jia told Yi the total points of his team, and Yi immediately knew the number of wins and losses of Jia's team in the entire tournament. What is the total score of Jia's team in this tournament?
10. Consider the case with $n$ teams.

Let the team where Jia is located win $x$ games, draw $y$ games, and lose $z$ games. Then the total score of the team is
$$
S=3 x+y \quad (x+y+z=n-1, x, y, z \geqslant 0).
$$

Consider the region $\Omega:\left\{\begin{array}{l}x+y \leqslant n-1, \\ x \geqslant 0, \\ y \geqslant 0 .\end{array}\right.$
When $S$ changes, the equation $S=3 x+y$ represents a set of parallel lines with a slope of -3. We need to determine the value of $S$ such that the line $l$ intersects the region $\Omega$ at exactly one integer point, and we call such a line $l$ a "good line."
Consider the intersection point $A(0, S)$ of the line $l$ with the $y$-axis.
When $S=0,1,2$, $l$ is a good line.
When $3 \leqslant S \leqslant n-1$, $l$ passes through at least two integer points in the region $\Omega$, so $l$ is not a good line.

When $S \geqslant n$, consider the intersection point $B\left(\frac{S}{3}, 0\right)$ of the line $l$ with the $x$-axis.

When $S=3 n-9$, $l$ passes through $B_{1}(n-3,0)$, and additionally, $l$ passes through the integer point $(n-4,3)$ in the region $\Omega$, so $l$ is not a good line.
When $S \leqslant 3 n-9$, the intersection point of $l$ with the $x$-axis lies on the segment
$O B_{1}$, and $l$ passes through at least two integer points in the region $\Omega$, so $l$ is not a good line.

When $S=3 n-4$, as shown in Figure 2, $l$ does not pass through any integer points in the region $\Omega$, so $l$ is not a good line.

When $3 n-8 \leqslant S \leqslant 3 n-5$ or $S=3 n-3$, as shown in Figure 2, $l$ passes through exactly one integer point in the region $\Omega$, so $l$ is a good line.

In summary, $S=0,1,2,3 n-8,3 n-7,3 n-6,3 n-5,3 n-3$.
Taking $n=2013$, we get
$$
S=0,1,2,6031,6032,6033,6034,6036 .
$$