
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 $f(x)=ax^2+bx+c$ $(a > 0,b,c\in\mathbb{R})$, and $\varphi(x)=\begin{cases}f(x) & ,x\geqslant 0 \\ -f(x) & ,x < 0\end{cases}$.  
$(1)$ If the minimum value of $f(x)$ is $f(-1)=0$, and $f(0)=1$, find the value of $\varphi(-1)+f(2)$;  
$(2)$ If $a=1,c=0$, and $|f(x)|\leqslant 1$ holds for all $x\in[0,1]$, find the range of $b$;  
$(3)$ If $a=1,b=-2,c=0$, and the graph of $y=\varphi(x)$ and $y=-t$ have exactly one common point in the closed interval $[-2,t]$, find the range of the real number $t$.

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