
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
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Which of the following propositions are correct?<br/>① If $a+b \gt 2c$, then $C＜\frac{π}{3}$;<br/>② If $a^{2}+b^{2} \gt c^{2}$, then $\triangle ABC$ is an acute triangle;<br/>③ If $cos^{2}\frac{A}{2}=\frac{b+c}{2c}$, then $\triangle ABC$ is a right triangle;<br/>④ If $\frac{a}{cosB}=\frac{b}{cosA}$, then $\triangle ABC$ is an isosceles triangle;<br/>⑤ If $\triangle ABC$ is an acute triangle, then $sinA+sinB+tanC＞\frac{1}{2}cosA+\frac{1}{3}cosB+\frac{1}{4}cotC$ always holds.

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