
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
Solve the following system of equations:

$$
\begin{aligned}
& \left|a_{1}-a_{2}\right| \cdot x_{2}+\left|a_{1}-a_{3}\right| \cdot x_{3}+\left|a_{1}-a_{4}\right| \cdot x_{4}=1 \\
& \left|a_{2}-a_{1}\right| \cdot x_{1} \quad+\left|a_{2}-a_{3}\right| \cdot x_{3}+\left|a_{2}-a_{4}\right| \cdot x_{4}=1 \\
& \left|a_{3}-a_{1}\right| \cdot x_{1}+\left|a_{3}-a_{2}\right| \cdot x_{2} \quad+\left|a_{3}-a_{4}\right| \cdot x_{4}=1 \\
& \left|a_{4}-a_{1}\right| \cdot x_{1}+\left|a_{4}-a_{2}\right| \cdot x_{2}+\left|a_{4}-a_{2}\right| \cdot x_{3} \quad=1
\end{aligned}
$$

where $a_{1}, a_{2}, a_{3}$, and $a_{4}$ are four distinct given real numbers.

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