
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 $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with the right vertex $A$ and the top vertex $B$, a line passing through points $A$ and $B$ divides the circumference of the circle ${(x-1)^2}+{({y-\frac{\sqrt{3}}{2}})^2}=1$ in half, and forms a triangle with the coordinate axes with an area of $\sqrt{3}$.<br/>$(1)$ Find the equation of the ellipse $E$;<br/>$(2)$ If the line $l:y=x+m$ intersects $E$ at points $C$ and $D$, with point $M(0,3m)$, find the equation of the line $l$ when the area of triangle $CDM$ is maximized.

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