
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
With inspiration drawn from the rectilinear network of streets in *New York* , the *Manhattan distance* between two points  $(a,b)$  and  $(c,d)$  in the plane is defined to be \[|a-c|+|b-d|\] Suppose only two distinct *Manhattan distance* occur between all pairs of distinct points of some point set. What is the maximal number of points in such a set?

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