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

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

(2) Let point $(a,b)$ be a random point in the region $\begin{cases} x+y-8\leqslant 0 \\ x > 0 \\ y > 0 \end{cases}$. Calculate the probability that the function $y=f(x)$ is an increasing function in the interval $[1,+\infty)$.

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