
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 parabola $\Gamma: x^2 = 4y$ and a point $H(1, -1)$, find the equation of the circle with the line segment $AB$ as its diameter, where $A$ and $B$ are the points of intersection between the parabola and the two tangent lines $HA$ and $HB$ passing through point $H$.

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