
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 $Γ:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ passing through the point $A(\sqrt{3},\frac{\sqrt{13}}{2})$, and the left focus of $Γ$ is $F(-2\sqrt{3},0)$, a line $l$ intersects $Γ$ at points $M$ and $N$.  
$(1)$ Find the equation of the ellipse $Γ$;  
$(2)$ If $\overrightarrow{MP}=\frac{1}{2}\overrightarrow{PN}$ and the coordinates of point $P$ are $\left(0,1\right)$, find the slope of the line $l$;  
$(3)$ If $|\overrightarrow{OM}+\overrightarrow{ON}|=4$, where $O$ is the origin, find the maximum area of triangle $MON$.

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