
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 all real values of \(a\) for which there exist five non-negative real numbers \(x_{1}, \ldots, x_{5}\) satisfying the following equations:

\[ 
\sum_{k=1}^{5} k x_{k} = a 
\]

\[ 
\sum_{k=1}^{5} k^{3} x_{k} = a^{2} 
\]

\[ 
\sum_{k=1}^{5} k^{5} x_{k} = a^{3} 
\]

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