
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 the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$ with its right focus at $F$, the top vertex at $M$, and the origin at $O$, if the area of $\triangle OMF$ is $\dfrac {1}{2}$, and the eccentricity of the ellipse is $\dfrac { \sqrt {2}}{2}$.

$(1)$ Find the equation of the ellipse;  
$(2)$ Determine whether there exists a line $l$ intersecting the ellipse at points $P$ and $Q$, such that point $F$ is the orthocenter of $\triangle PQM$. If it exists, find the equation of line $l$; if not, explain why.

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