
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 $\triangle ABC$, with $2B=A+C$, and the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, then among the following propositions, the correct ones are __________ (Fill in all the correct proposition numbers).

① $B=\dfrac{\pi }{3}$;

② If $\cos 2A+\cos 2B+\cos 2C > 1$, then $\triangle ABC$ is an obtuse triangle;

③ If $a=2c$, then $\triangle ABC$ is an acute triangle;

④ If ${ \overset{→}{AB}}^{2}= \overset{→}{AB}· \overset{→}{AC}+ \overset{→}{BA}· \overset{→}{CA}· \overset{→}{CB} $, then $3A=C$;

⑤ If $\tan A+\tan C+\sqrt{3} > 0$, then $\triangle ABC$ is an obtuse triangle;

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