
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 a quadratic function $f(x) = ax^2 - 4bx + 1$.

(1) Let set $P = \{-1,1,2,3,4,5\}$ and set $Q = \{-2,-1,1,2,3,4\}$. Randomly select a number from set $P$ as $a$ and from set $Q$ as $b$. Calculate the probability that the function $y = f(x)$ is increasing on the interval $[1,+\infty)$.

(2) Suppose the point $(a, b)$ is a random point within the region defined by $\begin{cases} x+y-8\leqslant 0, \\ x > 0, \\ y > 0 \end{cases}$. 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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