
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]


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Given an ellipse C: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b > 0$, let F<sub>1</sub> and F<sub>2</sub> be the left and right foci, respectively.
(I) If the sum of the distances from point A ($\sqrt{6}$, $\frac{2\sqrt{6}}{3}$) on ellipse C to foci F<sub>1</sub> and F<sub>2</sub> is equal to 6, write the equation of ellipse C and the coordinates of the foci;
(II) Suppose point K is a moving point on ellipse C obtained in (I), find the equation for the trajectory of the midpoint M of segment F<sub>1</sub>K.

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