
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
Among the following propositions, the correct ones are __________, (write down all the correct proposition numbers)

① If an odd function $f(x)$ has a period of $4$, then the graph of the function $f(x)$ is symmetric about $(2,0)$;  
② If $a\in(0,1)$, then $a^{1+a} < a^{1+ \frac {1}{a}}$;  
③ The function $f(x)=\ln \frac {1+x}{1-x}$ is an odd function;  
④ There exists a unique real number $a$ such that $f(x)=\lg (ax+ \sqrt {2x^{2}+1})$ is an odd function.

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