
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
Consider  a sphere  $S$  of radius   $R$  tangent to a plane  $\pi$ . Eight other spheres of the same radius  $r$  are tangent to  $\pi$  and tangent to  $S$ , they form a "collar" in which each is tangent to its two neighbors. Express  $r$  in terms of  $R$ .

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