
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
In a village of 2011 dwarfs, each dwarf is either wearing a red hat or a blue hat. Dwarfs with red hats always tell the truth, while dwarfs with blue hats can lie. Additionally, they can change the color of their hats at any time. One day, each dwarf encountered every other dwarf exactly once, and each one claimed that the other was wearing a blue hat. What is the minimum number of times the dwarfs changed the color of their hats that day?

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