
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 with a base side length of 4 and height of 2, first place a sphere \( B_1 \) that is tangent to the base and all three lateral faces. Then, for \( n = 2, 3, \cdots \), place sphere \( B_n \) in the pyramid such that it is externally tangent to sphere \( B_{n-1} \) and tangent to all three lateral faces of the pyramid. Find the total volume of all these spheres.

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