
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
Given that \( x_{1} \) and \( x_{2} \) are distinct complex numbers and satisfy the following three equations:
$$
\begin{array}{l}
x_{1}(x_{1}+1)=A, \quad x_{2}(x_{2}+1)=A, \\
x_{1}^{4}+3 x_{1}^{3}+5 x_{1}=x_{2}^{4}+3 x_{2}^{3}+5 x_{2}.
\end{array}
$$

Find the value of \( A \) that satisfies the above equations.

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