
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 $C$ with its center at the origin and its foci on the x-axis, one of its vertices $B$ has coordinates $(0, 1)$, and its eccentricity is $\frac{\sqrt{2}}{2}$. Line $l$ intersects ellipse $C$ at points $M$ and $N$.  
(Ⅰ) Find the equation of ellipse $C$;  
(Ⅱ) If the right focus $F$ of ellipse $C$ is exactly the orthocenter of $\triangle BMN$, find the equation of line $l$.

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