
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
Find all real numbers \( a \) such that there exist non-negative real numbers \( x_1, x_2, x_3, x_4, x_5 \) satisfying
\[
\sum_{k=1}^{5} k x_k = a, \quad \sum_{k=1}^{5} k^3 \cdot x_k = a^2, \quad \sum_{k=1}^{5} k^5 x_k = a^3.
\]

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