
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 numbers \( z_1, z_2, z_3 \) such that \( \frac{z_1}{z_2} \) is a purely imaginary number and
\[ \left|z_{1}\right| = \left|z_{2}\right| = 1, \left|z_{1} + z_{2} + z_{3}\right| = 1. \]
Then, the minimum value of \( \left|z_{3}\right| \) is \(\qquad\) .

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