
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
Given the geometric sequence $\left\{z_{n}\right\}$, if $z_{1}=1$, $z_{2}=a+b i$, and $z_{3}=b+a i$ where $a$ and $b$ are real numbers and $a>0$, find the smallest value of $n$ such that $z_{1}+z_{2}+\cdots+z_{n}=0$.

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