
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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