
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 following inequalities hold:
1. \( \cos A + \cos B + \cos C \leq \frac{3}{2} \);
2. \( \sin \frac{A}{2} \cdot \sin \frac{B}{2} \cdot \sin \frac{C}{2} \leq \frac{1}{8} \);
3. \( abc \geq 8(p-a)(p-b)(p-c) \);
4. \( R \geq 2r \);
5. \( S_{\triangle} \leq \frac{1}{2} Rp \).

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