
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
Solve the equation
\[
\left(x_{1}^{2}+1\right)\left(x_{2}^{2}+2^{2}\right) \cdots\left(x_{n}^{2}+n^{2}\right)=2^{n} \cdot n! x_{1} \cdot x_{2} \cdots x_{n},
\]
where \( n \in \mathbf{N}^{*} \) and \( x_{1}, x_{2}, \cdots, x_{n} \in \mathbf{R} \).

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