
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
Consider a $5\times5$ rectangular array of points, each of which is $1$ unit apart from its nearest neighbors in a horizontal or vertical direction. Define an increasing distance path as a sequence of distinct points such that each consecutive point-to-point distance in the path is greater than the previous one. Determine the maximum number of points (m) in such a path and the number of such paths (r) that include exactly m points. Calculate the product mr.

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