
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
Let \\(S_n\\) denote the sum of the first \\(n\\) terms of the sequence \\(\{a_n\}\\). It is known that \\(\frac{S_5}{S_{10}} = \frac{1}{3}\\). If \\(\{a_n\}\\) is a geometric sequence, then the common ratio \\(q= \) ; if \\(\{a_n\}\\) is an arithmetic sequence, then \\(\frac{S_{10}}{S_{20}}= \) .

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