
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 $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > b > 0$) with eccentricity $e = \frac{\sqrt{3}}{2}$, the right and top vertices are respectively $A$ and $B$. The chord length intercepted by the circle $O: x^2 + y^2 = 1$ on line $AB$ is $\frac{2\sqrt{5}}{5}$.

(1) Find the equation of ellipse $C$;  
(2) Let a line $l$ passing through point $B$ with slope $k$ intersect ellipse $C$ at another point $M$, and $\overrightarrow{ON} = \lambda (\overrightarrow{OB} + \overrightarrow{OM})$. If point $N$ is on circle $O$, find the range of values for the positive real number $\lambda$.

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