
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
Two circles  $\mathcal{C}_1$  and  $\mathcal{C}_2$  with centers  $(1, 1)$  and  $(4, 5)$  and radii  $r_1 < r_2$ , respectively, are drawn on the coordinate plane. The product of the slopes of the two common external tangents of  $\mathcal{C}_1$  and  $\mathcal{C}_2$  is  $3$ . If the value of  $(r_2 - r_1)^2$  can be expressed as a common fraction in the form  $\tfrac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers, find  $m + n$ . 

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