
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 \( \tau(n) \) be the number of positive divisors of \( n \). Let \( \tau_{1}(n) \) be the number of positive divisors of \( n \) which have remainders 1 when divided by 3. Find all possible integral values of the fraction \( \frac{\tau(10 n)}{\tau_{1}(10 n)} \).

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