
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
In a right triangular pyramid (not a regular tetrahedron), the area of the base is three times less than the area of a lateral face. The height of the pyramid is 10 cm.

Construct the following (infinite) sequence of spheres. Let $S_{1}$ be the inscribed sphere in this pyramid. Then $S_{2}$ is a sphere that touches the lateral faces of the pyramid and the sphere $S_{1}$; $S_{3}$ is a sphere that touches the lateral faces of the pyramid and the sphere $S_{2}$, not equal to $S_{1}$ and so on; $S_{n+1}$ is a sphere that touches the lateral faces of the pyramid and the sphere $S_{n}$, not equal to $S_{n-1}$.

Find the total volume of all these spheres.

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