
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 distinct complex numbers \( x_{1} \) and \( x_{2} \) such that the following three equations hold:
\[
\begin{aligned}
x_{1}(x_{1}+1) &= A, \\
x_{2}(x_{2}+1) &= A, \\
x_{1}^{4} + 3x_{1}^{3} + 5x_{1} &= x_{2}^{4} + 3x_{2}^{3} + 5x_{2}.
\end{aligned}
\]

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