
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
A circle is centered at point  $O$  in the plane. Distinct pairs of points  $A, B$  and  $C, D$  are diametrically opposite on this circle. Point  $P$  is chosen on line segment  $AD$  such that line  $BP$  hits the circle again at  $M$  and line  $AC$  at  $X$  such that  $M$  is the midpoint of  $PX$ . Now, the point  $Y \neq X$  is taken for  $BX = BY, CD \parallel XY$ . IF  $\angle PYB = 10^{\circ}$ , find the measure of  $\angle XCM$ .


Proposed by Albert Wang (awang11)

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