
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
For triangle $ABC$ with internal angles $A$, $B$, and $C$, and opposite sides $a$, $b$, and $c$ respectively, determine which of the following statements are true:

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. Given $\overrightarrow{a} = (\tan A + \tan B, \tan C)$ and $\overrightarrow{b} = (1, 1)$, if $\overrightarrow{a} \cdot \overrightarrow{b} > 0$, then triangle $ABC$ is an acute triangle.
4. If $O$ is the circumcenter of triangle $ABC$, then $\overrightarrow{AO} \cdot \overrightarrow{BC} = \frac{1}{2}(b^2 - c^2)$.
5. If $\sin^2 A + \sin^2 B = \sin^2 C$ and $\overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC} = \overrightarrow{0}$, then $\frac{|\overrightarrow{OA}|^2 + |\overrightarrow{OB}|^2}{|\overrightarrow{OC}|^2} = 5$.

Indicate the correct sequence number(s).

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