
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
In life, ellipses have many optical properties, such as light rays emanating from one focus of an ellipse and reflecting off the elliptical mirror surface, with the reflected light passing through the other focus. Now, let ellipse $C$ have its foci on the $y$-axis, with the center at the origin. A light ray emitted from the lower focus $F_{1}$ of the ellipse reflects off the elliptical mirror surface to the upper focus $F_{2}$. The total length of this light ray is $4$, and the maximum area of the triangle formed by the reflection point and the focus is $\sqrt{3}$. It is known that the eccentricity of the ellipse is $e＜\frac{\sqrt{2}}{2}$.  
$(1)$ Find the standard equation of ellipse $C$.  
$(2)$ If two light rays $OM$ and $ON$ are emitted from the center $O$ of ellipse $C$, passing through points $A$ and $B$ on the ellipse, and then reaching points $M$ and $N$ on the line $y=4$, with the line segment $AB$ passing through the upper focus $F_{2}$ of the ellipse, can the lines $BM$ and $AN$ intersect at a certain point? If so, find this point; if not, explain why.

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