
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
Let the function $y=f(x)$ have a domain $D$. If there exists a non-zero constant $T$, such that for any $x \in D$, $f(x+T)=T \cdot f(x)$ holds, then the function $y=f(x)$ is called a "pseudo-periodic function", and the non-zero constant $T$ is called the "pseudo-period" of the function $y=f(x)$. There are four propositions about "pseudo-periodic functions" as follows:  
1. If the "pseudo-period" of the "pseudo-periodic function" $y=f(x)$ is $-1$, then it is a periodic function with a period of $2$;  
2. The function $f(x)=x$ is a "pseudo-periodic function";  
3. The function $f(x)=2^{x}$ is a "pseudo-periodic function";  
4. If the function $f(x)=\cos \omega x$ is a "pseudo-periodic function", then "$\omega=k\pi$, $k \in Z$".  
Among these, the true propositions are \_\_\_\_\_\_ (Write down all the proposition numbers that satisfy the condition).

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