
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  $m>1$  be a fixed positive integer. For a nonempty string of base-ten digits  $S$ , let  $c(S)$  be the number of ways to split  $S$  into contiguous nonempty strings of digits such that the base-ten number represented by each string is divisible by  $m$ . These strings are allowed to have leading zeroes.

In terms of  $m$ , what are the possible values that  $c(S)$  can take?

For example, if  $m=2$ , then  $c(1234)=2$  as the splits  $1234$  and  $12|34$  are valid, while the other six splits are invalid.

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