
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
Let the number \( a \) have the property that for any four real numbers \( x_{1}, x_{2}, x_{3}, x_{4} \), one can always choose integers \( k_{1}, k_{2}, k_{3}, k_{4} \) such that
\[ 
\sum_{1 \leqslant i < j \leqslant 4} \left( \left( x_{j} - k_{i} \right) - \left( x_{i} - k_{j} \right) \right)^{2} \leqslant a . 
\]
Find the minimum value of such \( a \).

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