
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 function $f\left(x\right)=ax^{2}-bx-1$, sets $P=\{1,2,3,4\}$, $Q=\{2,4,6,8\}$, if a number $a$ and a number $b$ are randomly selected from sets $P$ and $Q$ respectively to form a pair $\left(a,b\right)$.<br/>$(1)$ Let event $A$ be "the monotonically increasing interval of the function $f\left(x\right)$ is $\left[1,+\infty \right)$", find the probability of event $A$;<br/>$(2)$ Let event $B$ be "the equation $|f\left(x\right)|=2$ has $4$ roots", find the probability of event $B$.

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