
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 \( z \) is a complex number, \( |z| = 1 \), and \( u = z^{4} - z^{3} - 3z^{2}i - z + 1 \). Find the maximum value of \( |u| \) and the complex number \( z \) when the maximum value is achieved.

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