
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, and an area of $S$, and an inscribed circle O with radius $r$. Connect OA, OB, and OC, then the areas of triangles OAB, OBC, and OAC are respectively $\frac{1}{2}cr$, $\frac{1}{2}ar$, and $\frac{1}{2}br$. From $S= \frac{1}{2}cr+ \frac{1}{2}ar+ \frac{1}{2}br$, we get $r= \frac{2S}{a+b+c}$. By analogy, for a tetrahedron with volume $V$ and the areas of its four faces being $S_1$, $S_2$, $S_3$, and $S_4$, the radius $R$ of the inscribed sphere is \_\_\_\_\_\_.

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