
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
Suppose that $x_{1}, \ldots, x_{n}$ are real numbers for which it is known that all numbers $\sigma_{1}=\sum x_{i}, \sigma_{2}=\sum_{i<j} x_{i} x_{j}, \sigma_{3}=\sum_{i<j<k} x_{i} x_{j} x_{k}, \ldots, \sigma_{n}=x_{1} \ldots x_{n}$ are positive. Is it true that the numbers $x_{1}, \ldots, x_{n}$ themselves are positive?

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