
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
Number  $a$  is such that  $\forall a_1, a_2, a_3, a_4 \in \mathbb{R}$ , there are integers  $k_1, k_2, k_3, k_4$  such that  $\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a$ . Find the minimum of  $a$ .

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