
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 \( n \geq 3 \) be an integer and \( x_{2}, \ldots, x_{n} \) be positive real numbers such that \( x_{2} \times \cdots \times x_{n} = 1 \). Show the following inequality:
$$
\left(1 + x_{2}\right)^{2} \times \left(1 + x_{3}\right)^{3} \times \cdots \times \left(1 + x_{n}\right)^{n} > n^{n}
$$

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