
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 diagram below shows a  $ 4\times4$  rectangular array of points, each of which is  $ 1$  unit away from its nearest neighbors.
[asy]unitsize(0.25inch);
defaultpen(linewidth(0.7));

int i, j;
for(i = 0; i < 4; ++i)
	for(j = 0; j < 4; ++j)
		dot(((real)i, (real)j));[/asy]Define a *growing path* to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let  $ m$  be the maximum possible number of points in a growing path, and let  $ r$  be the number of growing paths consisting of exactly  $ m$  points. Find  $ mr$ .

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