
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  $I$  be the incenter of a triangle  $ABC$  with  $AB = 20$ ,  $BC = 15$ , and  $BI = 12$ . Let  $CI$  intersect the circumcircle  $\omega_1$  of  $ABC$  at  $D \neq C $ . Alice draws a line  $l$  through  $D$  that intersects  $\omega_1$  on the minor arc  $AC$  at  $X$  and the circumcircle  $\omega_2$  of  $AIC$  at  $Y$  outside  $\omega_1$ . She notices that she can construct a right triangle with side lengths  $ID$ ,  $DX$ , and  $XY$ . Determine, with proof, the length of  $IY$ .

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