
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 an ellipse $C$: $\dfrac{y^2}{a^2}+\dfrac{x^2}{b^2}=1$ ($a > b > 0$) with eccentricity $\dfrac{1}{2}$, and its upper focus point is $F(0,1)$. A moving line $l$ passing through $F$ intersects the ellipse $C$ at points $M$ and $N$. Let point $P(3,4)$, and denote the slopes of $PM$ and $PN$ as $k_1$ and $k_2$, respectively.

(Ⅰ) Find the equation of the ellipse $C$;

(Ⅱ) If the slope of line $l$ is $-1$, find the value of $k_1·k_2$;

(Ⅲ) Investigate whether $\dfrac{1}{k_1}+\dfrac{1}{k_2}$ is a constant value? If yes, find this constant value; if not, find the range of $\dfrac{1}{k_1}+\dfrac{1}{k_2}$.

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