
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 that the complex numbers \( z_{1} \) and \( z_{2} \) satisfy \( \left| z_{1} \right| = \left| z_{1} + 2z_{2} \right| \) and \( \overline{z_{1}} z_{2} = a(2 - i) \) (where \( a \) is a non-zero real number), find \( \frac{z_{2}}{z_{1}} \).

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