
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 area $S$ and circumradius $R$. Perpendiculars are erected at vertices $A$, $B$, and $C$ to the plane of the triangle, and points $A_{1}$, $B_{1}$, and $C_{1}$ are taken on these perpendiculars such that segments $A A_{1}$, $B B_{1}$, and $C C_{1}$ have lengths equal to the respective altitudes of the triangle dropped from vertices $A$, $B$, and $C$. Find the volume of the pyramid bounded by the planes $A_{1}B_{1}C$, $A_{1}BC_{1}$, $AB_{1}C_{1}$, and $ABC$.

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