
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
The distance between two cells on an infinite chessboard is defined as the minimum number of moves a king needs to travel between these cells. Three cells on the board are marked, and the pairwise distances between them are all equal to 100. How many cells exist such that the distance from these cells to each of the three marked cells is 50?

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