
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, the left and right foci of the ellipse and one endpoint of its minor axis form the three vertices of an equilateral triangle. Point $D(1, \frac{3}{2})$ is on the ellipse $C$, and the line $l: y = kx + m$ intersects the ellipse $C$ at points $A$ and $P$, and intersects the $x$-axis and $y$-axis at points $N$ and $M$, respectively, with $|PM| = |MN|$. The point $Q$ is the reflection of point $P$ across the $x$-axis, and the extension of line $QM$ intersects the ellipse $C$ at point $B$. Perpendicular lines are drawn from points $A$ and $B$ to the $x$-axis, with the feet of the perpendiculars being $A_1$ and $B_1$, respectively.  
$(1)$ Find the equation of the ellipse $C$.  
$(2)$ Does there exist a line $l$ such that point $N$ bisects the line segment $A_1B_1$? If so, find the equation of line $l$; if not, explain why.

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