
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 $\dfrac{x^2}{2} + y^2 = 1$.

$(1)$ Find the equation of the trajectory of the midpoint of parallel chords with a slope of $2$;

$(2)$ A line $l$ passing through $A(2,1)$ intersects with the ellipse, find the equation of the trajectory of the midpoint of the chord defined by $l$;

$(3)$ Find the equation of the line that passes through point $P\left( \dfrac{1}{2}, \dfrac{1}{2} \right)$ and is bisected by point $P$.

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