
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 $\frac{x^2}{2} + y^2 = 1$, find:
1. The equation of the line containing the chord bisected at point $P(\frac{1}{2}, \frac{1}{2})$;
2. The trajectory equation of the midpoints of the chords with a slope of $2$;
3. The trajectory equation of the midpoints of the chords intercepted by the tangent line passing through point $A(2, 1)$;
4. The trajectory equation of the midpoints of the line segments $PQ$, where $P$ and $Q$ are points on the ellipse, and the product of the slopes of the lines $OP$ and $OQ$ is $-\frac{1}{2}$, with $O$ being the origin.

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