
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  $ABC$  be a triangle with  $AB=26$ ,  $AC=28$ ,  $BC=30$ . Let  $X$ ,  $Y$ ,  $Z$  be the midpoints of arcs  $BC$ ,  $CA$ ,  $AB$  (not containing the opposite vertices) respectively on the circumcircle of  $ABC$ . Let  $P$  be the midpoint of arc  $BC$  containing point  $A$ . Suppose lines  $BP$  and  $XZ$  meet at  $M$  , while lines  $CP$  and  $XY$  meet at  $N$ . Find the square of the distance from  $X$  to  $MN$ .

*Proposed by Michael Kural*

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