
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
The equation of the locus of a moving point \( z_1 \) in the complex plane is \( \left|z_1 - z_0 \right| = \left| z_1 \right| \), 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 point \( z \) and describe its shape and position in the complex plane.

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