
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 parabola $C$ is given by the equation $x^2=2py$ ($p>0$). For a point $Q(m, \frac{1}{2})$ on $C$, the distance to the focus is 1.

(Ⅰ) Determine the equation of the parabola $C$.

(Ⅱ) Let a line $l$ passing through point $M(0, 2)$ intersect the parabola $C$ at points $A$ and $B$, where the x-coordinate of point $A$ is $n$ ($n\in\mathbb{N}^*$):

(ⅰ) Denote the area of triangle $AOB$ as $f(n)$; find the expression of $f(n)$.

(ⅱ) Investigate whether there exists different points $A$ such that the areas of different triangles $AOB$ are equal. If such points exist, find the coordinates of point $A$; if not, provide a reason.

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