
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
In a 2-dimensional coordinate system \( xOy \), triangle \( \triangle ABC \) has vertices \( B(-1, 0) \), \( C(1, 0) \), and a moving point \( A \). Points \( H \) and \( G \) are the orthocenter and centroid of \( \triangle ABC \), respectively. If the midpoint \( K \) of line segment \( HG \) lies on line \( BC \):

1. Find the equation of the locus \( E \) of point \( A \);
2. A line passing through point \( C \) and with a slope of 1 intersects the curve \( E \) at point \( D \). If there exist two points \( P \) and \( Q \) on curve \( E \) such that \( P \), \( C \), \( Q \), and \( D \) are concyclic, find the minimum area of the circumscribed circle of quadrilateral \( PCQD \).

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