
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
Several sages stood in a column. They all wore either black or white hats. It turned out that among any 10 consecutive sages, there were an equal number of sages with white and black hats, but among any 12 consecutive sages, they were not equal. What is the maximum number of sages that could be in the column?

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