
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

Let \( P\left( x_{0}, y_{0} \right) \) be a point within the ellipse \( \Gamma \) (different from the center \( O \)). Draw a line \( l \) through point \( P \) which intersects the ellipse at points \( A \) and \( B \). Construct a parallelogram \( O A Q B \) with \( O A \) and \( O B \) as adjacent sides (in the degenerate case when \( A \), \( O \), and \( B \) are collinear). The locus of point \( Q \) is an ellipse \( \Gamma^{\prime} \) centered at \( P \) and having the same eccentricity as the original ellipse \( \Gamma \). Moreover, ellipse \( \Gamma^{\prime} \) passes through the center of \( \Gamma \).

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