
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
You are familiar with the rules of solitaire: there are marbles on a board, and you eliminate marbles by jumping over them with another marble, etc. Now consider the plane and place marbles on all the integer points of the negative half-plane. The goal of the game is to place a marble as high as possible in a finite number of moves. The diagram shows how to place a marble at height 1. What is the maximum height that can be reached?

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