
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 triangle $ABC$ with side lengths $a$, $b$, and $c$, area $S$, and an inscribed circle $O$ with radius $r$. If we connect $OA$, $OB$, and $OC$, the areas of triangles $OAB$, $OBC$, and $OAC$ are $\frac{1}{2}cr$, $\frac{1}{2}ar$, and $\frac{1}{2}br$, respectively. From $S=\frac{1}{2}cr+\frac{1}{2}ar+\frac{1}{2}br$, we get $r=\frac{2S}{a+b+c}$. By analogy, find the radius $R$ of the inscribed sphere of a tetrahedron with volume $V$ and surface areas $S_{1}$, $S_{2}$, $S_{3}$, and $S_{4}$.

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