
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
For a rational number  $r$ , its *period* is the length of the smallest repeating block in its decimal expansion. for example, the number  $r=0.123123123...$  has period  $3$ . If  $S$  denotes the set of all rational numbers of the form  $r=\overline{abcdefgh}$  having period  $8$ , find the sum of all elements in  $S$ .

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