
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 the conditions that the given numbers \(\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}\) must satisfy for the system of 6 equations with 4 unknowns

$$
\left\{\begin{array}{l}
x_{1}+x_{2}=\alpha_{1} \alpha_{2} \\
x_{1}+x_{3}=\alpha_{1} \alpha_{3} \\
x_{1}+x_{4}=\alpha_{1} \alpha_{4} \\
x_{2}+x_{3}=\alpha_{2} \alpha_{3} \\
x_{2}+x_{4}=\alpha_{2} \alpha_{4} \\
x_{3}+x_{4}=\alpha_{3} \alpha_{4}
\end{array}\right.
$$

to have a solution. Find the values of the unknowns \(x_{1}, x_{2}, x_{3}, x_{4}\).

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