
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
Find the greatest  $n$  such that  $(z+1)^n = z^n + 1$  has all its non-zero roots in the unitary circumference, e.g.  $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$  implies  $|\alpha| = 1.$ 

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