
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  $\Gamma_1$  be a circle with radius  $\frac{5}{2}$ .  $A$ ,  $B$ , and  $C$  are points on  $\Gamma_1$  such that  $\overline{AB} = 3$  and  $\overline{AC} = 5$ . Let  $\Gamma_2$  be a circle such that  $\Gamma_2$  is tangent to  $AB$  and  $BC$  at  $Q$  and  $R$ , and  $\Gamma_2$  is also internally tangent to  $\Gamma_1$  at  $P$ .  $\Gamma_2$  intersects  $AC$  at  $X$  and  $Y$ .  $[PXY]$  can be expressed as  $\frac{a\sqrt{b}}{c}$ . Find  $a+b+c$ .

*2022 CCA Math Bonanza Individual Round #5*

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