
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 shows a $3\times3$ rectangular array of points, where each point is $1$ unit away from its nearest vertical or horizontal neighbors.
[asy] 
unitsize(0.25inch); 
defaultpen(linewidth(0.7));  
int i, j; 
for(i = 0; i < 3; ++i) 
    for(j = 0; j < 3; ++j) 
        dot(((real)i, (real)j)); 
[/asy]
Define a growing path to be a sequence of distinct points in the array such that the sequence only involves vertical or horizontal moves and the distance between consecutive points of the sequence is strictly increasing. Let $m$ be the maximum possible number of points in such 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