
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
Let \(a, b\) be nonzero complex numbers, and \(\frac{a}{b}\) is not a real number. Define:
\[ 
L_{a, b}=\{r a+s b \mid r, s \in \mathbf{Z}\}, 
\]
\[ 
R_{a, b}=\left\{z \mid z \text{ is a nonzero complex number, and } L_{a, b} = L_{z a, z}\right\}.
\]

Find the maximum number of elements in the set \(R_{a, b}\) as \(a, b\) change.

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