
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
Garfield and Odie are situated at  $(0,0)$  and  $(25,0)$ , respectively. Suddenly, Garfield and Odie dash in the direction of the point  $(9, 12)$  at speeds of  $7$  and  $10$  units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as  $\frac{m}{\sqrt{n}}$  for relatively prime positive integers  $m$  and  $n$ . Find  $m+n$ .

*Proposed by **Th3Numb3rThr33***

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