
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 that points \( A, B, \) and \( C \) lie on a circle with center \( O \), and \( \overrightarrow{A O} = \frac{1}{2}(\overrightarrow{A B} + \overrightarrow{A C}) \), find the dot product \( \overrightarrow{A B} \cdot \overrightarrow{A C} \).

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