
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 set $A=\{x\in Z|x^{2}-\left(2t+1\right)x+4t-2 \lt 0\}$.  
$(1)$ If $1\notin A$, find the range of $t$.  
$(2)$ If the number of subsets of $A$ is $4$, is there a maximum value for $|t-2|$? If so, find the maximum value; if not, explain why.

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