
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
Consider a sequence of complex numbers \(\left\{z_{n}\right\}\) classified as "interesting" if \(\left|z_{1}\right|=1\) and for any positive integer \(n\), the relation \(4 z_{n+1}^{2}+2 z_{n} z_{n+1}+z_{n}^{2}=0\) holds. Determine the largest constant \(C\) such that for any interesting sequence \(\left\{z_{n}\right\}\) and any positive integer \(m\), the inequality \(\left| z_{1} + z_{2} + \cdots + z_{m} \right| \geq C\) is satisfied.

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