
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 the Cartesian coordinate system $xOy$, for the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, the left focus is $F(-1,0)$, the left vertex is $A$, and the top and bottom vertices are $B$ and $C$, respectively.  
$(1)$ If the line $BF$ passes through the midpoint $M$ of $AC$, find the standard equation of the ellipse $E$;  
$(2)$ If the slope of line $BF$ is $1$, and the other intersection point of $BF$ with the ellipse is $D$, find the area of $\triangle BCD$.

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