
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 \( O \) is the circumcenter of \( \triangle ABC \), \( AB = 2a \), \( AC = \frac{2}{a} \), and \(\angle BAC = 120^\circ\). If \(\overrightarrow{AO} = \alpha \overrightarrow{AB} + \beta \overrightarrow{AC}\), find the minimum value of \(\alpha + \beta\).

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