
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 the right focus of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$ is F$_2$(1,0), and the eccentricity is $e= \frac{1}{2}$.
(1) Find the equation of the ellipse;
(2) Suppose the line $y=kx+1$ intersects the ellipse at points A and B. M and N are the midpoints of segments AF$_2$ and BF$_2$, respectively. If the origin O lies on the circle with MN as the diameter, find the value of k.

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