You are a helpful assistant that can solve the given question step by step with the help of the wikipedia search tool and python interpreter tool. Given a question, you need to first think about the reasoning process in the mind and then provide the answer. During thinking, you can invoke the wikipedia search tool to search and python interpreter tool to calculate the math problem for fact information about specific topics if needed. The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags respectively, and the search query and result are enclosed within <search> </search> and <result> </result> tags respectively. For example, <think> This is the reasoning process. </think> <search> search query here </search> <result> search result here </result> <think> This is the reasoning process. </think> <python> python code here </python> <result> python interpreter result here </result> <think> This is the reasoning process. </think> <answer> The final answer is \[ oxed{answer here} \] </answer>. In the last part of the answer, the final exact answer is enclosed within oxed{} with latex format.
system
Twelve participants are taking part in a chess tournament. After the tournament, each participant makes 12 lists as follows: 
- In the 1st list, each participant writes down only their own name.
- In the 2nd list, each participant writes down their own name and the names of all the opponents they defeated.
- In the 3rd list, each participant writes down all the names from their 2nd list as well as the names of all the opponents defeated by anyone in the 2nd list.
- This process continues until the 12th list, where each participant writes down all the names from their 11th list and the names of all the opponents defeated by anyone in the 11th list.

It is known that for each participant, their 12th list contains names that did not appear in their 11th list. How many draws were there in the tournament?
user
54
rule
Twelve participants are taking part in a chess tournament. After the tournament, each participant makes 12 lists as follows: 
- In the 1st list, each participant writes down only their own name.
- In the 2nd list, each participant writes down their own name and the names of all the opponents they defeated.
- In the 3rd list, each participant writes down all the names from their 2nd list as well as the names of all the opponents defeated by anyone in the 2nd list.
- This process continues until the 12th list, where each participant writes down all the names from their 11th list and the names of all the opponents defeated by anyone in the 11th list.

It is known that for each participant, their 12th list contains names that did not appear in their 11th list. How many draws were there in the tournament?
train