
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 \(a\) for which there exist non-negative real numbers \(x_1, x_2, x_3, x_4, x_5\) satisfying:
\[x_1 + 2x_2 + 3x_3 + 4x_4 + 5x_5 = a,\]
\[x_1 + 2^3x_2 + 3^3x_3 + 4^3x_4 + 5^3x_5 = a^2,\]
\[x_1 + 2^5x_2 + 3^5x_3 + 4^5x_4 + 5^5x_5 = a^3.\]

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