
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  $\triangle ABC$ ,  $AB=17$ ,  $AC=25$ , and  $BC=28$ .  Points  $M$  and  $N$  are the midpoints of  $\overline{AB}$  and  $\overline{AC}$  respectively, and  $P$  is a point on  $\overline{BC}$ .  Let  $Q$  be the second intersection point of the circumcircles of  $\triangle BMP$  and  $\triangle CNP$ .  It is known that as  $P$  moves along  $\overline{BC}$ , line  $PQ$  passes through some fixed point  $X$ .  Compute the sum of the squares of the distances from  $X$  to each of  $A$ ,  $B$ , and  $C$ .

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