
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

There are 10 players \( A_{1}, A_{2}, \cdots, A_{10} \), with initial scores of \( 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 \) respectively. These players will compete in a round-robin tournament where each pair of players competes exactly once, and each match must have a decisive outcome.

If the higher-ranked player wins, the winner gets 1 point and the loser gets 0 points. If the lower-ranked player wins, the winner gets 2 points and the loser gets 0 points. After all matches are completed, the cumulative scores (initial scores plus points earned in the tournament) are calculated for each player and the players are reordered based on these cumulative scores.

What is the minimum cumulative score of the new champion (the player with the highest cumulative score after all matches)?

Present the answer in LaTex format: \boxed{Your answer}
user