
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
How are the decimal representations of the fractions $\left\{\frac{p}{q}\right\}$ and $\left\{\frac{10^{k} p}{q}\right\}$ related?

Find the possible values of the denominator of a fraction of the form $\left\{\frac{1}{a}\right\}$, which represents a purely periodic decimal fraction with two digits in the period.

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