
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 an ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 (a > b > 0)$, its left and right foci are $F_1$ and $F_2$ respectively. A point $P(1,\dfrac{3}{2})$ on the ellipse and the line connecting $P$ to the right focus of the ellipse is perpendicular to the x-axis.

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

$(2)$ A line $l$ that is tangent to the parabola $y^2=4x$ in the first quadrant intersects the ellipse $C$ at points $A$ and $B$, and intersects the x-axis at point $M$. The perpendicular bisector of segment $AB$ intersects the y-axis at point $N$. Find the minimum value of the slope of line $MN$.

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