
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 triangle  $ABC$  with incenter  $I$  and circumcircle  $\Gamma$  satisfy  $AB = 6\sqrt{3}, BC = 14,$  and  $CA = 22$ . Construct points  $P$  and  $Q$  on rays  $BA$  and  $CA$  such that  $BP = CQ = 14$ . Lines  $PI$  and  $QI$  meet the tangents from  $B$  and  $C$  to  $\Gamma$ , respectively, at points  $X$  and  $Y$ . If  $XY$  can be expressed as  $a\sqrt{b} - c$  for positive integers  $a,b,c$  with  $c$  squarefree, find  $a + b + c$ .

*Proposed by Andrew Wu*

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