
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
The Co-circular Points Theorem states that the following statements are equivalent, meaning that each implies the others.

(1) Points \( A, B, C, D \) are co-circular (lie on the same circle).

(2) There exists a point \( O \) such that \( OA = OB = OC = OD \).

(3) If points \( C \) and \( D \) are on the same side of line \( AB \), then \(\angle ACB = \angle ADB\).

(4) If points \( C \) and \( D \) are on the same side of line \( AB \), then \( \angle BAD + \angle DCB = \pi \).

(5) If points \( C \) and \( D \) are on the same side of line \( AB \), then \( \angle BAD = \pi - \angle DCB \).

(6) If lines \( AC \) and \( BD \) intersect at point \( P \), then \( PA \cdot PC = PB \cdot PD \).

(7) If lines \( AB \) and \( DC \) intersect at point \( Q \), then \( QA \cdot QB = QC \cdot QD \).

(8) \( AB \cdot DC + AD \cdot BC = AC \cdot BD \).

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