
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 \( 12 \mid 34 \) are valid.

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