
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: Extracting several terms (at least three) from a sequence $\{a_n\}$ and arranging them in the order they appear in $\{a_n\}$ creates a subsequence of $\{a_n\}$. If the subsequence forms an arithmetic (or geometric) progression, it is called an arithmetic (or geometric) subsequence of $\{a_n\}$.
(1) Find the geometric subsequences of the sequence $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$.
(2) Let $\{a_n\}$ be a geometric sequence where each term is a real number and the common ratio $q \ne 1$.
(i) Provide an example of $\{a_n\}$ which contains an infinite arithmetic subsequence (the process need not be shown).
(ii) If $\{a_n\}$ contains an infinite arithmetic subsequence, find all possible values of $q$.

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