
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
Eight positive integers are written on the faces of a cube, such that each face shares an edge with exactly three other faces. Each vertex is labeled with the product of the four numbers on the faces adjacent to the vertex (including faces that share only a vertex). If the sum of the numbers on the vertices is equal to $2002$, then what is the sum of the numbers written on the faces?

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