
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 $y=x^{2}-\left(a+2\right)x+4\left(a\in R\right)$.<br/>$(1)$ If the solution set of the inequality $y \lt 0$ with respect to $x$ is $\{x\left|\right.1 \lt x \lt b\}\left(b \gt 1\right)$, find the values of $a$ and $b$;<br/>$(2)$ If $y\geqslant -a-1$ holds for all $1\leqslant x\leqslant 4$, find the range of real number $a$.

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