
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
Determine the smallest positive integer $n$ such that there are real numbers $x_1, x_2, \dots, x_n$ satisfying:
\begin{align*}
x_1 + x_2 + x_3 + \dots + x_n &= 500, \\
x_1^4 + x_2^4 + x_3^4 + \dots + x_n^4 &= 256000.
\end{align*}

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