
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
Given the following propositions, the correct ones are \_\_\_\_\_\_ (write down all the correct proposition numbers)

① The graph of the function $f(x)=\log_{a}(x-3)+2$ always passes through the fixed point $(4,2)$;  
② Given the sets $P=\{a,b\}$, $Q=\{0,1\}$, then there are $1$ mappings $f$: $P\rightarrow Q$ that satisfy $f(b)=0$;  
③ If the range of the function $f(x)=\log_{2}(x^{2}-2ax+1)$ is $\mathbb{R}$, then the range of the real number $a$ is $(-1,1)$;  
④ The equation of the function symmetric to $f(x)=e^{x}$ about $y=x$ is $y=\ln x$.

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