
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
Which of the functions

$$
\begin{aligned}
& \varphi(t)=\exp \left(-|t|^{\alpha}\right), \quad \alpha \in(0,2], \\
& \varphi(t)=\exp \left(-|t|^{\alpha}\right), \quad \alpha>2, \\
& \varphi(t)=\frac{1}{\left(1+|t|^{\alpha}\right)^{\beta}}, \quad \alpha \in(0,2], \beta>0 \\
& \varphi(t)=\frac{1}{\left(1+|t|^{\alpha}\right)^{\beta}}, \quad \alpha>2, \beta>0 \\
& \varphi(t)=\left\{\begin{array}{ll}
(1-|t|)^{\beta}, & |t| \leqslant 1, \\
0, & |t|>1,
\end{array} \quad \beta \geqslant 1,\right. \\
& \varphi(t)=\left\{\begin{array}{ll}
(1-|t|)^{\beta}, & |t| \leqslant 1, \\
0, & |t|>1,
\end{array} \quad \beta \in(0,1)\right. \\
& \varphi(t)= \begin{cases}1-|t|^{3}, & |t| \leqslant 1 \\
0, & |t|>1,\end{cases} \\
& \varphi(t)= \begin{cases}1-|t|, & |t| \leqslant 1 / 2 \\
1 /(4|t|), & |t|>1 / 2\end{cases}
\end{aligned}
$$

are characteristic functions?

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