
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 $f(x) = ax^2 - 4bx + 1$ with respect to $x$.
(I) Let set $P = \{1, 2, 3\}$ and set $Q = \{-1, 1, 2, 3, 4\}$. A number is randomly selected from set $P$ as $a$ and another number is randomly selected from set $Q$ as $b$. Find the probability that the function $f(x)$ is increasing in the interval $[1, +\infty)$.
(II) Let point $(a, b)$ be a random point within the region $\begin{cases} x + y - 8 \leqslant 0 \\ x > 0 \\ y > 0 \end{cases}$. Find the probability that the function $f(x)$ is increasing in the interval $[1, +\infty)$.

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