
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
Let $F_1$ and $F_2$ be 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 $A(1, \frac{3}{2})$ on ellipse $C$ to points $F_1$ and $F_2$ equals 4, write the equation of ellipse $C$ and the coordinates of the foci;  
(2) Let point $K$ be a moving point on the ellipse obtained in (1), find the equation of the trajectory of the midpoint of segment $F_1K$.

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