
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
A fair six-faced cubic die, labeled with numbers 0, 1, 2, 3, 4, 5, and a fair four-faced tetrahedral die, labeled with numbers 1, 2, 3, 4, are rolled once simultaneously. Let $z = a + bi$ be a complex number, where $a$ is the number on the face-up side of the cube, and $b$ is the sum of the numbers on the three visible sides of the tetrahedron.

(1) List the set $A = \{z \mid z \text{ is a pure imaginary number}\}$.

(2) Calculate the probability that the point $(a, b)$ in the complex plane corresponding to the complex number $z$ satisfies the inequality ${a}^{2} + {\left(b-6\right)}^{2} \leqslant 9$.

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