
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 the complex numbers \( z_{1}, z_{2} \) correspond to the points \( A, B \) respectively in the complex plane, and \( \left|z_{2}\right| = 4 \) and \( 4z_{1}^2 - 2z_{1}z_{2} + z_{2}^2 = 0 \), determine the area of the triangle \( AOB \).

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