
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
1. Does there exist an infinite sequence of positive integers \(\{a_{n}\}\) such that for any positive integer \(n\), the following inequality holds?
$$
a_{n+1}^{2} \geqslant 2 a_{n} a_{n+2}
$$

2. Does there exist an infinite sequence of positive irrational numbers \(\{a_{n}\}\) such that for any positive integer \(n\), the following inequality holds?
$$
a_{n+1}^{2} \geqslant 2 a_{n} a_{n+2}
$$

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