
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
If \( z_{1} \) and \( z_{2} \) are two non-zero complex numbers with equal modulus, then the necessary and sufficient condition for \( \left(z_{1} + z_{2}\right)^{2} = \lambda^{2} z_{1} z_{2} \) is \( \left|z_{1} + z_{2}\right| = \lambda \left|z_{k}\right| \) where \( \lambda \in \mathbf{R}^{+} \) and \( k = 1, 2 \).

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