
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
Triangle  $ABC$  with  $AB=4$ ,  $BC=5$ ,  $CA=6$  has circumcircle  $\Omega$  and incircle  $\omega$ . Let  $\Gamma$  be the circle tangent to  $\Omega$  and the sides  $AB$ ,  $BC$ , and let  $X=\Gamma \cap \Omega$ . Let  $Y$ ,  $Z$  be distinct points on  $\Omega$  such that  $XY$ ,  $YZ$  are tangent to  $\omega$ . Find  $YZ^2$ .
*The following fact may be useful: if  $\triangle{ABC}$  has incircle  $w$  with incenter  $I$  and radius  $r$ , and  $\triangle{DEF}$  is the intouch triangle (i.e.  $D$ ,  $E$ ,  $F$  are intersections of incircle with  $BC$ ,  $CA$ ,  $AB$ , respectively) and  $H$  is the orthocenter of  $\triangle{DEF}$ , then the inversion of  $X$  about  $\omega$  (i.e. the point  $X'$  on ray  $IX$  such that  $IX' \cdot IX=r^2$ ) is the midpoint of  $DH$ .*

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