
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 $f(x)=x^{2}-4x+a+3$, where $a\in\mathbb{R}$;  
$(1)$ If the function $y=f(x)$ has a zero point in the interval $[-1,1]$, find the range of values for $a$;  
$(2)$ Let the function $g(x)=bx+5-2b$, where $b\in\mathbb{R}$, when $a=3$, if for any $x_{1}\in[1,4]$, there always exists $x_{2}\in[1,4]$ such that $g(x_{1})=f(x_{2})$, find the range of values for $b$.

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