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
Let  $M$  be a set of six distinct positive integers whose sum is  $60$ . These numbers are written on the faces of a cube, one number to each face. A *move* consists of choosing three faces of the cube that share a common vertex and adding  $1$  to the numbers on those faces. Determine the number of sets  $M$  for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.
user
84
rule
Let  $M$  be a set of six distinct positive integers whose sum is  $60$ . These numbers are written on the faces of a cube, one number to each face. A *move* consists of choosing three faces of the cube that share a common vertex and adding  $1$  to the numbers on those faces. Determine the number of sets  $M$  for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.
train