
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
Call a complex number sequence $\{z_n\}$ "interesting" if $\left|z_1\right|=1$, and for any positive integer $n$, the equation $4z_{n+1}^2 + 2z_n z_{n+1} + z_n^2 = 0$ holds. Find the largest constant $C$ such that for all interesting sequences $\{z_n\}$ and any positive integer $m$, the inequality $\left|z_1 + z_2 + \cdots + z_m\right| \geq C$ holds.

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