
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
An ellipse $E$ is given by the equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity $e = \frac{1}{2}$. The distance $d$ between the right focus of the ellipse and the line $\frac{x}{a} + \frac{y}{b} = 1$ is $\frac{\sqrt{21}}{7}$. $O$ is the origin of the coordinate system.

(1) Find the equation of the ellipse $E$;

(2) Draw two perpendicular rays passing through point $O$, intersecting the ellipse $E$ at points $A$ and $B$ respectively. Find the distance between point $O$ and line $AB$.

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