
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 line \( l \) passing through \( A(2,1) \) intersects the hyperbola \( x^2 - \frac{y^2}{2} = 1 \) at two points \( P_1 \) and \( P_2 \). Find the equation of the locus of the midpoint of the line segment \( P_1 P_2 \).

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