
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 $ABC$, the sides opposite to internal angles $A$, $B$, $C$ are $a$, $b$, $c$, respectively. Then:
1. If $\cos B \cos C > \sin B \sin C$, then $\triangle ABC$ must be an obtuse triangle;
2. If $a\cos A = b\cos B$, then $\triangle ABC$ is an isosceles triangle;
3. If $\vec{a} = (\tan A + \tan B, \tan C)$ and $\vec{b} = (1, 1)$, and if $\vec{a} \cdot \vec{b} > 0$, then $\triangle ABC$ is an acute triangle;
4. If $O$ is the circumcenter of $\triangle ABC$, then $\vec{AO} \cdot \vec{BC} = \frac{1}{2}(b^2 - c^2)$;
5. If $\sin^2 A + \sin^2 B = \sin^2 C$ and $\vec{OA} + \vec{OB} + \vec{OC} = \vec{0}$, then $\frac{|\vec{OA}|^2 + |\vec{OB}|^2}{|\vec{OC}|^2} = 5$.

The correct statements are __________ .

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