
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
The vertices of the convex quadrilateral  $ABCD$  lie on the parabola  $y=x^2$ . It is known that  $ABCD$  is cyclic and  $AC$  is a diameter of its circumcircle. Let  $M$  and  $N$  be the midpoints of the diagonals of  $AC$  and  $BD$  respectively. Find the length of the projection of the segment  $MN$  on the axis  $Oy$ . 

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