
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 

\[ 
|a-c| + |b-d| \text{.}
\]

Suppose only two distinct Manhattan distances 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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