
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 focus $F$, the upper vertex $B$, $|BF|=2$, and eccentricity $\frac{1}{2}$.  
$(1)$ Find the standard equation of the ellipse $E$;  
$(2)$ If the line $l:y=x-2m(m\neq 0)$ intersects the ellipse $E$ at points $A$ and $C$, with point $N(0,m)$, find the equation of the line $l$ when the area of triangle $ACN$ is maximized.

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