
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 that the quadratic function $f(x)$ has a minimum value of $1$, and $f(0)=f(2)=3$.

$(1)$ Find the expression for $f(x)$;  
$(2)$ If $f(x)$ is monotonic on the interval $[2t,t+1]$, find the range of the real number $t$;  
$(3)$ Let $g(x)=f(x)+4(1-m)x$, for any real numbers $x_{1}$, $x_{2} \in [-1,1]$, it always holds that $|g(x_{1})-g(x_{2})| \leqslant 8$, find the range of the real number $m$.

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