
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  $n \geqslant 2$  be an integer. There are  $n$  finite sets  ${A_1},{A_2},\ldots,{A_n}$  which satisfy the condition 
\[\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \quad \forall i,j \in \left\{ {1,2,...,n} \right\}.\]
Find the minimum of  $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|} $ .

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