
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 $x_1,$ $x_2,$ $\dots,$ $x_n$ such that
\[
x_1 + x_2 + x_3 + \dots + x_n = 100,
\]
\[
x_1^3 + x_2^3 + x_3^3 + \dots + x_n^3 = 10000.
\]
Find the smallest positive integer $n$ for which this is possible.

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