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

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