
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
It is known that $F(0,1)$ is a focus of the ellipse $C$ with its center at the origin $O$, and the eccentricity $e$ of the ellipse $C$ is $\frac{1}{2}$.
(Ⅰ) Find the equation of the ellipse $C$;
(Ⅱ) Suppose $M(x_1,y_1)$ and $N(x_2,y_2)$ are two distinct points on the ellipse $C$, and the slope of line $MN$ is $k_1$; Point $A$ satisfies $\overrightarrow{OM} + \overrightarrow{ON} = \lambda \overrightarrow{OA}$ ($\lambda \neq 0$), and the slope of line $OA$ is $k_2$.
① Find the value of $k_1 \cdot k_2$;
② If the coordinates of $A$ are $(\frac{3}{2}, 1)$, find the range of the real number $\lambda$.

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