
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 regular triangular pyramid (not a regular tetrahedron), the area of the base is half the area of a lateral face. The height of the pyramid is 70 cm.

Consider the following (infinite) sequence of spheres. Let \( S_1 \) be the inscribed sphere of this pyramid. Then \( S_2 \) is the sphere which touches the lateral faces of the pyramid and the sphere \( S_1 \); \( S_3 \) is the sphere which touches the lateral faces of the pyramid and the sphere \( S_2 \), differing from \( S_1 \), and so forth; \( S_{n+1} \) is the sphere which touches the lateral faces of the pyramid and the sphere \( S_n \), differing from \( S_{n-1} \).

Find the total volume of all these spheres.

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