
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 $f(x)=x^{2}+ax+2$, where $a\in \mathbb{R}$.  
$(1)$ If for any $x\in [-1,1]$, the inequality $f(x)\leq 2a(x-1)+4$ always holds, find the range of real number $a$;  
$(2)$ Given $g(x)=-x+m$, when $a=-3$, if for any $x_{1}\in [1,4]$, there always exists $x_{2}\in (1,8)$ such that $f(x_{1})=g(x_{2})$, find the range of real number $m$.

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