
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 homogenous cube has its six faces labeled with the numbers: -2, -1, 0, 1, 2, 3. When tossed twice, the numbers appearing on the top faces are \( a \) and \( b \), respectively. Calculate the probability that the function \( f(x) = ax^2 + b\ln{x} \) is monotonically increasing.

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