
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 equations: \(|a_i - a_1| x_1 + |a_i - a_2| x_2 + |a_i - a_3| x_3 + |a_i - a_4| x_4 = 1\) for \(i = 1, 2, 3, 4\), where \(a_i\) are distinct real numbers.

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