
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 right focus of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ is $F(1,0)$, $M$ is the top point of the ellipse, $O$ is the origin, and $\triangle OMF$ is an isosceles right triangle.<br/>$(Ⅰ)$ Find the equation of the ellipse;<br/>$(Ⅱ)$ Determine whether there exists a line $l$ that intersects the ellipse at points $P$ and $Q$, such that point $F$ is the orthocenter of $\triangle PQM$ (the intersection point of the altitudes of the triangle). If it exists, find the equation of the line $l$; if not, explain the reason.

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