
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 faces of a hexahedron and an octahedron are equilateral triangles with side length \(a\). The ratio of the radii of the inscribed spheres of these two polyhedra is a simplified fraction \(\frac{m}{n}\). What is the product \(m \cdot n\)?

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