
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 five propositions:
① The solution set of the inequality $x^2-4ax+3a^2<0$ is $\{x|a<x<3a\}$;
② If the function $y=f(x+1)$ is an even function, then the graph of $y=f(x)$ is symmetric about $x=1$;
③ If the solution set of the inequality $|x-4|+|x-3|<a$ is empty, then it must be that $a\geq1$;
④ The graph of the function $y=f(x)$ has at most one intersection point with the line $x=a$;
⑤ If angles $\alpha$ and $\beta$ satisfy $\cos\alpha \cdot \cos\beta=1$, then $\sin(\alpha+\beta)=0$.
Among these, the correct propositions are numbered as follows.

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