
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
When 
$$
x_{1}^{x_{1}} x_{2} x_{2} \ldots x_{n}^{x_{n}} \quad \text{and} \quad x_{1} = x_{2} = \cdots = x_{n} = \frac{1}{n},
$$
find the minimum value.

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