Team Game

Two teams — Blue and Red — are playing an interesting team game. At the start of the game, the host selects three numbers: $n$, $m$, and $k$. Then $n$ players from the Blue team (occupying positions 1, 2, \ldots, n) and $m$ players from the Red team (occupying positions n+1, n+2, \ldots, n+m) stand in a circle.

Starting from the first Blue player, the host counts 1, 2, \ldots, k. When the $k$-th player is reached, they are eliminated from the game. Counting then resumes from the next player, again counting 1, 2, \ldots, k, and the $k$-th player is also eliminated. After every two eliminations, the second eliminated player is replaced by a new player.

To determine the team of the new player, the host looks at the teams of the two just eliminated players. If both players are from the same team, the host invites a Blue player. If the players are from different teams, the host invites a Red player. The process repeats (two eliminations, then one replacement), always starting from the player after the newly added one. With each step, the total number of players decreases by one (two eliminations and one replacement), so after n + m - 1 steps, only one player remains.

The team of the last remaining player wins the grand prize! Given $n$, $m$, and $k$, determine which team will win.

For example, for n = m = 3 and k = 2 (where "S" denotes a Blue player and "R" denotes a Red player; subscripts indicate the order the player entered):

1. Initial arrangement: S_1\ S_2\ S_3\ R_4\ R_5\ R_6. Counting begins at S_1. First elimination: S_2. Second elimination: R_4 (replaced by R_7). 2. Circle: S_1\ S_3\ R_7\ R_5\ R_6. Counting begins at R_5. First elimination: R_6. Second elimination: S_3 (replaced by R_8). 3. Circle: S_1\ R_8\ R_7\ R_5. Counting begins at R_7. First elimination: R_5. Second elimination: R_8 (replaced by S_9). 4. Circle: S_1\ S_9\ R_7. Counting begins at R_7. First elimination: S_1. Second elimination: R_7 (replaced by R_{10}). 5. Circle: S_9\ R_{10}. Counting begins at S_9. First elimination: R_{10}. Second elimination: S_9 (replaced by R_{11}). 6. Final arrangement: R_{11}