
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
The Tennis Federation has assigned qualification numbers to all its players: the strongest player has number one, the next strongest has number two, and so on. It is known that in matches between players whose qualification numbers differ by more than 2, the player with the lower number always wins. A tournament involving the top 1024 players is conducted in a single-elimination format: players are paired randomly in each round, and the winner of each match advances to the next round, halving the number of participants each round. Therefore, after the tenth round, the winner will be determined. What is the highest number the winner can have?

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