
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
The ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \left(a > b > 0\right)$ passes through the point $P\left(1, \dfrac{3}{2}\right)$, with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity $e= \dfrac{1}{2}$. $M$ and $N$ are two moving points on the line $x=4$, and $\overrightarrow{F_1M} \cdot \overrightarrow{F_2N}=0$.

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

$(2)$ Find the minimum value of $|MN|$;

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