
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
$174 S-A B C D-S^{\prime}$ is a regular octahedron with edge length 1. Sphere $O$ is outside this octahedron, tangent to face $S A B$, and tangent to the extended planes of faces $S B C$, $S A D$, and $S^{\prime} A B$. Find the radius $r$ of sphere $O$.

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