
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
Given a cube with its eight vertices labeled with the numbers $1, 2, 3, \cdots, 8$ in any order, define the number on each edge as $|i-j|$, where $i$ and $j$ are the labels of the edge’s endpoints. Let $S$ be the sum of the numbers on all the edges. Find the minimum value of $S$.

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