
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
Arrange arbitrarily  $1,2,\ldots ,25$  on a circumference. We consider all  $25$  sums obtained by adding  $5$  consecutive numbers. If the number of distinct residues of those sums modulo  $5$  is  $d$    $(0\le d\le 5)$ ,find all possible values of  $d$ .

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