
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 the complex number \( z \) that satisfies the condition that the absolute value of one of the arguments of \( \frac{z+1}{z+2} \) is \( \frac{\pi}{6} \), find the locus of the point \( Z \) corresponding to the complex number \( z \) and determine the range of the principal value of the argument of the complex number \( z \).

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