
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 squares of an infinite chessboard is defined as the minimum number of moves a king needs to travel between these squares. Three squares are marked on the board, where each pair of distances between them equals 100. How many squares are there from which the distance to all three marked squares is 50?

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