
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
Consider sets $P=\{2, 3\}$ and $Q=\{-1, 1, 2, 3, 4\}$. Randomly select one number from set P as $a$ and one number from set Q as $b$ to form the ordered pair $(a, b)$, and construct the function $f(x) = ax^2 - 6bx + 1$.

(1) List all possible ordered pairs $(a, b)$, and calculate the probability that $a \geq 2$ and $b \leq 3$.

(2) Determine the probability that the function $f(x)$ is increasing on the interval $[1, +\infty)$.

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