
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 $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$ with left and right vertices $A$ and $B$, and a focal distance of $2$. Point $P$ lies on the ellipse and is different from points $A$ and $B$. The product of the slopes of lines $PA$ and $PB$ is $-\frac{3}{4}$.
$(1)$ Find the standard equation of the ellipse $C$.
$(2)$ A line passing through point $F(-1,0)$, not coinciding with the $x$-axis, intersects the ellipse $C$ at points $M$ and $N$. The equation of the line $m$ is $x=-2a$. A line $ME$ is drawn perpendicular to line $m$ passing through point $M$ and intersecting $m$ at point $E$. Find the maximum area of triangle $OEN$.

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