
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 that the upper vertex of the ellipse $\frac{x^{2}}{2}+y^{2}=1$ is $B$ and the right focus is $F$, the line $l$ intersects the ellipse at two points $M$ and $N$. Determine whether there exists a line $l$ such that the point $F$ is (1) the centroid; (2) the orthocenter of triangle $BMN$. If such a line exists, find the equation of the corresponding line $l$; if not, provide an explanation.

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