
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
a) In a sports competition, there are 20 athletes and 9 judges. Each judge ranks the athletes from 1 to 20 based on their performances. The judges’ rankings for each athlete do not differ by more than three positions between any two judges. The final ranking for each athlete is based on their "average rank," which is the sum of the positions assigned by all 9 judges divided by the number of judges (9). What is the highest possible average rank for the top athlete among the 20 participants?

b) The tennis federation assigns ranking numbers to all tennis players in the country: number 1 to the strongest, number 2 to the next strongest, and so on. In any match between two players whose ranking numbers differ by more than two, the player with the lower number always wins. There are 1024 tennis players in the country, and they have a knockout tournament where all losers are eliminated in each round, and the remaining players are paired randomly for the next round. What is the highest possible ranking number for the tournament winner?

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