
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 the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ has its right focus at $F_2(3,0)$, and its eccentricity is $e=\dfrac{\sqrt{3}}{2}$.

$(1)$ Find the equation of the ellipse;

$(2)$ Suppose the line $y=kx$ intersects the ellipse at points $A$ and $B$, and $M$, $N$ are the midpoints of segments $AF_2$ and $BF_2$ respectively,

if the circle with diameter $MN$ passes through the origin $O$, find the value of $k$.

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