
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
João wrote some non-zero real numbers, all distinct, on the board such that if we take any one of them and square it, the result is greater than the product of any other two numbers written on the board.

(a) Explain why there cannot be three positive real numbers written on the board.

(b) Explain why there cannot be three negative real numbers written on the board.

(c) Explain why there cannot be two positive numbers and two negative numbers written on the board.

(d) Determine the largest possible number of numbers written on the board with an example of a set of numbers that satisfies the given condition.

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