
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 inequality about $x$: $|x^{2}+ax+b|\leqslant 2|x-4|\cdot |x+2|$ holds for any real number $x$.  
$(1)$ Find all values of real numbers $a$ and $b$ that satisfy the condition.  
$(2)$ If $x^{2}+ax+b\geqslant \left(m+2\right)x-m-15$ holds for $x \gt 1$, find the range of real number $m$.

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