
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
At a joint conference of the parties of liars and truth-tellers, a presidium of 32 people was elected and seated in four rows of 8 people each. During the break, each member of the presidium claimed that among their neighbors there are representatives of both parties. It is known that liars always lie and truth-tellers always tell the truth. What is the minimum number of liars necessary in the presidium for the described situation to be possible? (Two members of the presidium are neighbors if one is to the left, right, in front, or behind the other.)

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