
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
Write a number on each card: +1 or -1. You can point to 3 cards and ask: What is the product of the numbers on these three cards? (But you won't be told what the number on each card is). When there are (1) 30 cards; (2) 31 cards; (3) 32 cards, what is the minimum number of questions needed to determine the product of the numbers on all the cards? (4) In a circle, with 50 numbers: +1 or -1, if you can ask a question to know the product of the 3 consecutive numbers, what is the minimum number of questions needed to determine the product of the numbers on all 50 cards?

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