
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, the locus equation of the moving point \( Z_{1} \) is given by \(\left|z_{1}-z_{0}\right|=\left|z_{1}\right|\), where \( z_{0} \) is a fixed point and \( z_{0} \neq 0 \). Another moving point \( z \) satisfies \( z_{1} z = -1 \). Find the locus of the point \( z \) and describe its shape and position in the complex plane.

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