
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 that $F(0,1)$ is a focus of the ellipse $C$ centered 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) Let $M(x\_1,y\_1)$, $N(x\_2,y\_2)$ be distinct points on the ellipse $C$, the slope of the line $MN$ is $k\_1$; $A$ is a point satisfying $\overrightarrow{OM}+\overrightarrow{ON}=\lambda\overrightarrow{OA}$ ($\lambda\neq0$), and the slope of the line $OA$ is $k\_2$.

$\quad$(1) Find the value of $k\_1\cdot k\_2$;

$\quad$(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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