
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
Find all solutions in real numbers \(x_1, x_2, \ldots, x_{n+1}\), all at least 1, such that:

1. \( x_1^{1/2} + x_2^{1/3} + x_3^{1/4} + \cdots + x_n^{1/(n+1)} = n \cdot x_{n+1}^{1/2} \)
2. \( \frac{x_1 + x_2 + \cdots + x_n}{n} = x_{n+1} \)

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