
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
In triangle  $ABC$ ,  $AB=3\sqrt{30}-\sqrt{10}$ ,  $BC=12$ , and  $CA=3\sqrt{30}+\sqrt{10}$ . Let  $M$  be the midpoint of  $AB$  and  $N$  be the midpoint of  $AC$ . Denote  $l$  as the line passing through the circumcenter  $O$  and orthocenter  $H$  of  $ABC$ , and let  $E$  and  $F$  be the feet of the perpendiculars from  $B$  and  $C$  to  $l$ , respectively. Let  $l'$  be the reflection of  $l$  in  $BC$  such that  $l'$  intersects lines  $AE$  and  $AF$  at  $P$  and  $Q$ , respectively. Let lines  $BP$  and  $CQ$  intersect at  $K$ .  $X$ ,  $Y$ , and  $Z$  are the reflections of  $K$  over the perpendicular bisectors of sides  $BC$ ,  $CA$ , and  $AB$ , respectively, and  $R$  and  $S$  are the midpoints of  $XY$  and  $XZ$ , respectively. If lines  $MR$  and  $NS$  intersect at  $T$ , then the length of  $OT$  can be expressed in the form  $\frac{p}{q}$  for relatively prime positive integers  $p$  and  $q$ . Find  $100p+q$ .

*Proposed by Vincent Huang and James Lin*

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