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

(1) Find the equation of the ellipse $C$;
(2) A line $l$ is tangent to the parabola $y^{2}=4x$ in the first quadrant, intersecting the ellipse $C$ at points $A$ and $B$, and intersecting the $x$-axis at point $M$. The perpendicular bisector of line 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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