
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 real number \( a \) have the following property: for any four real numbers \( x_1, x_2, x_3, x_4 \), there always exist integers \( k_1, k_2, k_3, k_4 \) such that
$$
\sum_{1 \leqslant i < j \leqslant 4} \left[ \left( x_i - k_i \right) - \left( x_j - k_j \right) \right]^2 \leqslant a.
$$

Find the minimum value of such \( a \).

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