
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 die twice, observe the numbers that appear, and let the first number be $a$ and the second number be $b$.
(1) Calculate the probability that $a+b$ is divisible by 3.
(2) Calculate the probability that the equation $x^2-ax+b=0$ has a solution.
(3) Calculate the probability that the system of equations $$\begin{cases} x+by=3 \\ 2x+ay=2\end{cases}$$ has only positive solutions.

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