
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
In a geometric sequence \(\{z_{n}\}\), given \(z_{1}=1\), \(z_{2}=a+b \mathrm{i}\), \(z_{3}=b+a \mathrm{i}\) where \(a, b \in \mathbf{R}\) and \(a>0\), find the smallest value of \(n\) such that \(z_{1}+z_{2}+\cdots+z_{n}=0\), and compute the value of \(z_{1} z_{2} \cdots z_{n}\).

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