
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
For the function $y=f(x)$, if there exists $x_{0} \in D$ such that $f(-x_{0})+f(x_{0})=0$, then the function $f(x)$ is called a "sub-odd function" and $x_{0}$ is called a "sub-odd point" of the function. Consider the following propositions:  
$(1)$ Odd functions are necessarily "sub-odd functions";  
$(2)$ There exists an even function that is a "sub-odd function";  
$(3)$ If the function $f(x)=\sin (x+ \frac {\pi}{5})$ is a "sub-odd function", then all "sub-odd points" of this function are $\frac {k\pi}{2} (k\in \mathbb{Z})$;  
$(4)$ If the function $f(x)=\lg \frac {a+x}{1-x}$ is a "sub-odd function", then $a=\pm1$;  
$(5)$ If the function $f(x)=4^{x}-m\cdot 2^{x+1}$ is a "sub-odd function", then $m\geqslant \frac {1}{2}$. Among these, the correct propositions are ______. (Write down the numbers of all propositions you think are correct)

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