
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]


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Given that \(a\) and \(b\) are distinct positive numbers, two sequences are inserted between them: \(x_{1}, x_{2}, \cdots, x_{n}\) and \(y_{1}, y_{2}, \cdots, y_{n}\), such that \(a, x_{1}, x_{2}, \cdots, x_{n}, b\) form an arithmetic sequence and \(a, y_{1}, y_{2}, \cdots, y_{n}, b\) form a geometric sequence. Which of the following inequalities hold:

1. \(\frac{1}{n} \sum_{k=1}^{n} x_{k} > \sqrt{ab} + \left(\frac{\sqrt{a} - \sqrt{b}}{2}\right)^{2}\)
2. \(\frac{1}{n} \sum_{k=1}^{n} x_{k} > \frac{a + b}{2}\)
3. \(\sqrt[n]{y_{1} y_{2} \cdots y_{n}} < \sqrt{ab}\)
4. \(\sqrt[n]{y_{1} y_{2} \cdots y_{n}} < \frac{a + b}{2} - \left( \frac{\sqrt{a} - \sqrt{b}}{2} \right)^{2}\)

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