
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
Let  $w_{1}$  and  $w_{2}$  denote the circles  $x^{2}+y^{2}+10x-24y-87=0$  and  $x^{2}+y^{2}-10x-24y+153=0$ , respectively. Let  $m$  be the smallest positive value of  $a$  for which the line  $y=ax$  contains the center of a circle that is externally tangent to  $w_{2}$  and internally tangent to  $w_{1}$ . Given that  $m^{2}=p/q$ , where  $p$  and  $q$  are relatively prime integers, find  $p+q$ .

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