
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 the Cartesian coordinate system xOy, F is the focus of the parabola C: $x^2=2py$ ($p>0$), M is any point on parabola C in the first quadrant. A circle through points M, F, and O has its center at Q. The distance from point Q to the directrix of parabola C is $\frac{3}{4}$.
(Ⅰ) Find the equation of parabola C.
(Ⅱ) Does there exist a point M such that the line MQ is tangent to parabola C at point M? If it exists, find the coordinates of point M; if not, explain why not.
(Ⅲ) If the x-coordinate of point M is $\sqrt{2}$, and the line $l: y=kx+\frac{1}{4}$ intersects with parabola C at two distinct points A and B, and intersects the circle Q at two distinct points D and E, find the minimum value of $|AB|^2+|DE|^2$ when $\frac{1}{2} \leq k \leq 2$.

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