
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
Let \( n \) be a natural number and \( a_{1}, a_{2}, \ldots, a_{2n} \) be mutually distinct integers. Find all integers \( x \) satisfying
$$
\left(x - a_{1}\right) \cdot \left(x - a_{2}\right) \cdots \left(x - a_{2n}\right) = (-1)^{n} (n!)^{2}.
$$

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