
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 a triangle $ABC$, where angles $A$, $B$, and $C$ are opposite sides $a$, $b$, and $c$, respectively, and $a=6$, $4\sin B=5\sin C$. Consider the four statements below:
① The maximum area of $\triangle ABC$ is $40$.
② $\triangle ABC$ under the given conditions cannot be a right-angled triangle.
③ When $A=2C$, the perimeter of $\triangle ABC$ is $15$.
④ When $A=2C$, if $O$ is the incenter of $\triangle ABC$, then the area of $\triangle AOB$ is $\sqrt{7}$.
Determine which of these statements are correct (Write down all the correct statement numbers).

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