
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
The centers of the faces of a cube form the vertices of a regular octahedron. The orthocenters of the faces of this octahedron form the vertices of another cube, and this process continues infinitely. What is the sum of the volumes of all these cubes if the edge length of the first cube is $a$?

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