
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
Given the quadratic function $f(x) = ax^2 - bx + 1$, let the sets $P = \{1, 2, 3\}$ and $Q = \{-1, 1, 2, 3, 4\}$. Randomly select a number $a$ from set $P$ and a number $b$ from set $Q$ to form a pair $(a, b)$.
(1) List all the pairs $(a, b)$ and calculate the probability that the function $y = f(x)$ has a zero point.
(2) Calculate the probability that the function $y = f(x)$ is increasing on the interval $[1, +\infty)$.

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