
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
A fly is sitting on an edge (not at a vertex) of a cube with edge length \(a\). It is looking for the shortest path that traverses the six faces of the cube and returns to the starting point. How does the length of this path depend on the position of the starting point? What is the length of the shortest path?

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