
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
Throw a uniform cubic die (with faces numbered 1, 2, 3, 4, 5, 6) twice in succession, and let the number appearing the first time be $a$, and the number appearing the second time be $b$. Let the complex number $z=a+bi$.  
(1) Find the probability that the event "$z-3i$ is a real number" occurs;  
(2) Find the probability that the event "the corresponding point $(a,b)$ of the complex number $z$ in the complex plane satisfies $(a-2)^2+b^2\leq9$" occurs.

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