
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
Does there exist an infinite increasing sequence of integers in which every element with an odd index (starting from the third) is the arithmetic mean, and every element with an even index is the geometric mean of its neighboring elements?

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