
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
Two circles \( C \) and \( C' \) meet at points \( X \) and \( Y \). Find four points such that for any circle that touches \( C \) and \( C' \) at points \( P \) and \( Q \) respectively, and intersects the line \( XY \) at points \( R \) and \( S \), each of the lines \( PR \), \( PS \), \( QR \), and \( QS \) passes through one of these four points.

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