
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 $\left\{z_{n}\right\}$ defined as "interesting" if $\left|z_{1}\right|=1$ and for every positive integer $n$, the following holds:
$$
4 z_{n+1}^{2}+2 z_{n} z_{n+1}+z_{n}^{2}=0.
$$

Find the largest constant $C$ such that for any "interesting" sequence $\left\{z_{n}\right\}$ and any positive integer $m$, the inequality below holds:
$$
\left|z_{1}+z_{2}+\cdots+z_{m}\right| \geqslant C.
$$

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