
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 contestants \(A_{1}, A_{2}, \cdots, A_{10}\), with their scores being \(9, 8, 7, 6, 5, 4, 3, 2, 1, 0\) respectively, and their ranks being \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10\). Now, a round-robin competition is conducted where each pair of contestants competes exactly once, and each match results in a winner. If a contestant with a higher current rank wins against a contestant with a lower rank, the winner gets 1 point, and the loser gets 0 points. If a contestant with a lower current rank wins against a contestant with a higher rank, the winner gets 2 points, and the loser gets 0 points. After all matches are completed, the cumulative scores of each contestant (current score plus points earned in the competition) are calculated, and a new ranking is determined based on these cumulative scores. Determine the minimum possible cumulative score of the new champion (ties in ranking are allowed).

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