
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, list the correct ones (write down all the correct proposition numbers).
① The graph of the function $f(x) = \ln(x - 1) + 2$ always passes through the fixed point $(1, 2)$.
② If the domain of the function $f(x)$ is $[-1, 1]$, then the domain of the function $f(2x - 1)$ is $[-3, 1]$.
③ Given the sets $P = \{a, b\}$, $Q = \{-1, 0, 1\}$, then there are 3 mappings from $f: P \rightarrow Q$ that satisfy $f(b) = 0$.
④ If the domain of the function $f(x) = \log_2(x^2 - 2ax + 1)$ is $\mathbb{R}$, then the range of the real number $a$ is $(-1, 1)$.
⑤ The equation of the function $f(x) = e^x$ that is symmetric about the line $y = x$ is $y = \lg x$.

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