
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
A function $y=f(x)$ is considered a "quasi-odd function" if it satisfies $f(a+x)+f(a-x)=2b$ where $a$ and $b$ are not both zero. The point $(a,b)$ is referred to as the "central point" of the function $f(x)$. Consider the following propositions:
① The function $f(x)=\sin x + 1$ is a quasi-odd function;
② If a quasi-odd function $y=f(x)$ has a central point on $\mathbb{R}$ at $(a, f(a))$, then the function $F(x)=f(x+a)-f(a)$ is an odd function on $\mathbb{R}$;
③ Given that the function $f(x)=\sin(2x- \frac{\pi}{3})+2$ is a quasi-odd function, its central point is $(\frac{\pi}{3}+k\pi, 2)$;
④ Given that the function $f(x)=x^3-3x^2+6x-2$ is a quasi-odd function, its central point is $(1, 2)$;
Identify all correct propositions by listing their numbers.

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