
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
 $N$  teams take part in a league. Every team plays every other team exactly once during the league, and receives 2 points for each win, 1 point for each draw, and 0 points for each loss. At the end of the league, the sequence of total points in descending order  $\mathcal{A} = (a_1 \ge a_2 \ge \cdots \ge a_N )$  is known, as well as which team obtained which score. Find the number of sequences  $\mathcal{A}$  such that the outcome of all matches is uniquely determined by this information.

*Proposed by Dominic Yeo, United Kingdom.*

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