
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
Person A and Person B use spheres with a radius of 2 to build pyramids. Person A builds a square pyramid with \( n \) layers: the top layer has 1 sphere, the second layer has 4 spheres, the third layer has 9 spheres, and so on, with each sphere tangent to four spheres in the layer below. Person B builds a triangular pyramid with \( n \) layers: the top layer has 1 sphere, the second layer has 3 spheres, the third layer has 6 spheres, and so on, with each sphere tangent to three spheres in the layer below. If Person A uses 2109 spheres, how many spheres must Person B use to ensure that their pyramid is at least as tall as Person A's?

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