
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 sphere of radius $r$, an octahedron is circumscribed around it. Then, another sphere is inscribed inside this octahedron such that it passes through the vertices of the octahedron. An icosahedron is circumscribed around this new sphere, and the process continues by drawing subsequent regular polyhedra in the following order: octahedron, icosahedron, dodecahedron, tetrahedron, hexahedron. How do the radii of the spheres relate to each other as this process continues?

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