
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 $f(x) = ax^2 - bx + 1$ concerning $x$, let set $P = \{1, 2, 3\}$ and set $Q = \{-1, 1, 2, 3, 4\}$. Randomly take one number from set P and one from set Q as the values of $a$ and $b$, respectively.  
(1) Find the probability that the function $y = f(x)$ has a real zero.  
(2) Find the probability that the function $y = f(x)$ is an increasing function on the interval $[1, +\infty)$.

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