
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 $F_1$ and $F_2$ are the left and right foci of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, respectively.
1) If the sum of the distances from point $A(\sqrt{6}, \frac{2\sqrt{6}}{3})$ on ellipse $C$ to $F_1$ and $F_2$ is equal to $6$, write the equation of the ellipse $C$ and the coordinates of its foci.
2) Let point $K$ be a moving point on the ellipse obtained in part (1). Find the trajectory equation of the midpoint $M$ of segment $F_1K$.

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