
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through point $A(2,3)$, and its eccentricity $e = \frac{1}{2}$.

$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ Determine whether there exists a line $l$ passing through point $B(0,-4)$ that intersects the ellipse at two distinct points $M$ and $N$, satisfying $\overrightarrow{OM} \cdot \overrightarrow{ON} = \frac{16}{7}$ (where point $O$ is the origin). If such a line exists, find the equation of line $l$; if not, explain why.

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