
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 $(xOy)$, an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ has its left focus at $F(-1, 0)$, left vertex at $A$, and upper and lower vertices at $B$ and $C$ respectively.
1. If line $BF$ passes through the midpoint $M$ of $AC$, find the standard equation of ellipse $E$.
2. If the slope of line $BF$ is $1$, and $BF$ intersects the ellipse at another point $D$, find the distance from point $D$ to the right directrix of ellipse $E$.

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