
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 the 9-element set \( A = \{a + b \mathrm{i} \mid a, b \in \{1, 2, 3\}\} \), where \( \mathrm{i} \) is the imaginary unit. \( \alpha = \left(z_1, z_2, \cdots, z_9\right) \) is a permutation of all elements of \( A \), satisfying \( |z_1| \leq |z_2| \leq \cdots \leq |z_9| \). How many such permutations \( \alpha \) are there?

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