
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
For a certain positive integer \( n \), there exist real numbers \( x_1, x_2, \ldots, x_n \) such that
\[
x_1 + x_2 + x_3 + \dots + x_n = 800,
\]
\[
x_1^4 + x_2^4 + x_3^4 + \dots + x_n^4 = 409600.
\]
Find the smallest positive integer \( n \) for which this is possible.

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