
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 \( p \in \mathbb{R} \). In the complex plane, consider the equation
\[ x^2 - 2x + 2 = 0, \]
whose two complex roots are represented by points \( A \) and \( B \). Also consider the equation
\[ x^2 + 2px - 1 = 0, \]
whose two complex roots are represented by points \( C \) and \( D \). If the four points \( A \), \( B \), \( C \), and \( D \) are concyclic, find the value of \( p \).

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