
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
Given a curve $C$ where the distance from a point on the curve to the point $F(0,1)$ is less than its distance to the line $y=-3$ by $2$.

(Ⅰ) Find the equation of curve $C$;

(Ⅱ) A line $l$ passing through point $F$ with a slope of $k$ intersects curve $C$ at points $A$ and $B$, and intersects the circle $F: x^2 + (y-1)^2 = 1$ at points $M$ and $N$ (with $A$ and $M$ being adjacent).

(ⅰ) If $\overrightarrow{BF} = \lambda \overrightarrow{BA}$, when $\lambda \in \left[ \frac{1}{2}, \frac{2}{3} \right]$, find the range of values for $k$;

(ⅱ) Tangents to curve $C$ at points $A$ and $B$, denoted as $l_1$ and $l_2$ respectively, intersect at point $P$. Find the minimum value of the product of the areas of triangles $AMP$ and $BNP$.

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