
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  $x_1$ ,  $x_2$  such that the following three equations hold:
\begin{align*}x_1(x_1+1)&=Ax_2(x_2+1)&=Ax_1^4+3x_1^3+5x_1&=x_2^4+3x_2^3+5x_2.\end{align*}

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