
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 $f(x) = ax^2 - 4bx + 1$,
(Ⅰ) Suppose set $P = \{1, 2, 3\}$, and set $Q = \{-1, 1, 2, 3, 4\}$. Randomly choose a number as $a$ from set P and another number as $b$ from set Q. Calculate the probability that function $f(x)$ is increasing on the interval $[1, +\infty)$.
(Ⅱ) Suppose the point $(a, b)$ is a random point within the region defined by $\begin{cases} x+y-8 \leq 0 \\ x > 0 \\ y > 0 \end{cases}$. Calculate the probability that function $f(x)$ is increasing on the interval $[1, +\infty)$.

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