
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
For which $n$ do there exist positive numbers $x_{1}, x_{2}, \ldots, x_{n}$ satisfying the equations

$$
x_{1} + x_{2} + \ldots + x_{n} = 3, \quad \frac{1}{x_{1}} + \frac{1}{x_{2}} + \ldots + \frac{1}{x_{n}} = 3?
$$

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