
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 that \(\lambda_{i} \in \mathbf{R}_{+}(i=1,2, \cdots, n)\), find all positive real solutions \(x_{1}, x_{2}, \cdots, x_{n}\) to the equation
$$
\sum_{i=1}^{n} \frac{\left(x_{i}+\lambda_{i}\right)^{2}}{x_{i+1}}=4 \sum_{i=1}^{n} \lambda_{i}
$$
(where \(x_{n+1}=x_{1}\)).

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