
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
Given the following propositions:
1. If k > 0, then the equation x^2 - 2x - k = 0 has real roots;
2. If x + y ≠ 8, then x ≠ 2 or y ≠ 6;
3. The false proposition of "If xy = 0, then at least one of x and y is 0" is \_\_\_\_\_\_.

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