
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
A tetrahedron P-ABC has its four vertices on the same spherical surface, with PA, PB, and PC being mutually perpendicular. The areas of the three lateral faces of the tetrahedron are $\sqrt{2}$, $2\sqrt{3}$, and $\sqrt{6}$, respectively. The radius of this sphere is ______.

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