
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 $M$ with the right focus at $F(1,0)$ and passing through the point $D(1, \frac{3}{2})$, which is defined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$:

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

$(2)$ A line $l$ passing through $F$ intersects the ellipse $M$ at points $A$ and $B$ (not coinciding with point $D$). The lines $DA$ and $DB$ intersect the $x$-axis at points $M$ and $N$, respectively. Is there a line $l$ such that $\angle DMN = \angle DNM$? 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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