
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 functions $f(x)=x^{2}-(a-2)x+4$ and $g(x)=\frac{x+b-3}{a{x}^{2}+2}$.  
$(1)$ If the function $f(x)$ is an even function on $[-b, b^{2}-b-3]$, find the value of the real number $b$.  
$(2)$ Under the condition in $(1)$, when the domain of $g(x)$ is $(-1,1)$, answer the following two questions:  
① Determine the monotonicity of the function $g(x)$ on its domain and provide a proof.  
② If $g(t-1)+g(2t) \lt 0$, find the range of the real number $t$.

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