
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
Definition: A subsequence of a sequence $\{a\_n\}$ is a sequence formed by extracting several terms ($\\geq 3$) from $\{a\_n\}$ in the order they appear. A subsequence with equal differences between terms is called an arithmetic subsequence of $\{a\_n\}$.

1. Find the arithmetic subsequences of the sequence $1$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\frac{1}{5}$.
2. Suppose that $\{a\_n\}$ is a geometric sequence of real numbers with common ratio $q \neq 1$.
   i. Give an example of $\{a\_n\}$ that has an infinite arithmetic subsequence.
   ii. If $\{a\_n\}$ has an infinite arithmetic subsequence, find all possible values of $q$.

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