
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$, where the left and right vertices are $A_1$ and $A_2$, the top and bottom vertices are $B_2/B_1$, and the left and right foci are $F_1$ and $F_2$, respectively. The major axis is $4$, and the circle $O$: $x^2 + y^2 = \dfrac{12}{7}$ is inscribed in the rhombus $A_1B_1A_2B_2$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Let point $N(n,0)$ be a point on the positive half of the $x$-axis, and a tangent line $l$ to the ellipse $C$ passes through point $N$. Let $H$ be the projection of the right focus $F_2$ on $l$. If the area of $\triangle F_1HN$ is not less than $\dfrac{3}{16}n^2$, find the range of $n$.

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