
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
A complex number sequence \(\{z_{n}\}\) is called "interesting" if \(|z_{1}| = 1\), and for any positive integer \(n\), \(4 z_{n+1}^{2} + 2 z_{n} z_{n+1} + z_{n}^{2} = 0\). Find the largest constant \(C\) such that for all interesting sequences \(\{z_{n}\}\) and positive integers \(m\), \(|z_{1} + z_{2} + \cdots + z_{m}| \geq C\).

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