
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
In the complex plane, let the vertices \( A \) and \( B \) of triangle \( \triangle AOB \) correspond to the complex numbers \( \alpha \) and \( \beta \), respectively, and satisfy the conditions: \( \beta = (1 + i)\alpha \) and \( |\alpha - 2| = 1 \). \( O \) is the origin. Find the maximum value of the area \( S \) of the triangle \( \triangle OAB \).

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