
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
The giants have taken over the kingdom of dwarves. The giants decided to deal with the dwarves and said the following to them: "Tomorrow we will line you up so that each of you will see those who are standing behind and will not see those who are standing in front (i.e., the first one sees everyone, the last one sees no one). Each of you will be given either a black or white hat (with equal probability, each will have either a black or a white hat) and will be asked what color it is. Those who answer correctly will be released, and those who answer incorrectly will be executed." What is the minimum number of dwarves at risk when certain agreements are made before the execution, given that there are \( p \) dwarves in total and \( p < \infty \)? Justify your answer. (12 points)

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