
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
If $\varphi(t)$ is the characteristic function of some random variable $\xi$, will $\operatorname{Re} \varphi(t)$, $\operatorname{Im} \varphi(t)$, $\overline{\varphi(t)}$, $|\varphi(t)|$, and $|\varphi(t)|^{2}$ also be characteristic functions?

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