
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
Circles  $\mathcal{C}_1$ ,  $\mathcal{C}_2$ , and  $\mathcal{C}_3$  have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line  $t_1$  is a common internal tangent to  $\mathcal{C}_1$  and  $\mathcal{C}_2$  and has a positive slope, and line  $t_2$  is a common internal tangent to  $\mathcal{C}_2$  and  $\mathcal{C}_3$  and has a negative slope. Given that lines  $t_1$  and  $t_2$  intersect at  $(x,y)$ , and that  $x=p-q\sqrt{r}$ , where  $p$ ,  $q$ , and  $r$  are positive integers and  $r$  is not divisible by the square of any prime, find  $p+q+r$ .

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