
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
One hundred sages want to travel by electric train consisting of 12 wagons from the first to the 76th station. They know that at the first station, two controllers will board the train into two different wagons. After the fourth station, at each interval, one of the controllers will move to a neighboring wagon, taking turns. A sage can only see a controller if they are in an adjacent or next-to-adjacent wagon. At each station, each sage can change wagons by running on the platform, but they can move at most three wagons away (for example, from the 7th wagon, a sage can move to any wagon numbered from 4 to 10). What is the maximum number of sages that can avoid ever being in the same wagon as a controller, regardless of how the controllers move? (The sages receive no information about the controllers apart from what's stated in the problem, and they must agree on a strategy in advance.)

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