
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$), respectively. Suppose point $A(1, \dfrac{3}{2})$ is on the ellipse $C$, and $F_1(-1,0)$, $F_2(1,0)$,

(1) Write the standard equation of the ellipse $C$;  
(2) Let $K$ be a moving point on the ellipse, find the equation of the trajectory of the midpoint $B$ of the line segment $KF_1$;

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