
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
Bill draws two circles which intersect at  $X,Y$ . Let  $P$  be the intersection of the common tangents to the two circles and let  $Q$  be a point on the line segment connecting the centers of the two circles such that lines  $PX$  and  $QX$  are perpendicular. Given that the radii of the two circles are  $3,4$  and the distance between the centers of these two circles is  $5$ , then the largest distance from  $Q$  to any point on either of the circles can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $100m+n$ .

*Proposed by Tristan Shin*

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