
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
Find, with proof, all pairs of positive integers  $(n,d)$  with the following property: for every integer  $S$ , there exists a unique non-decreasing sequence of  $n$  integers  $a_1,a_2,...,a_n$  such that  $a_1 + a_2 + ... + a_n = S$  and  $a_n-a_1=d.$ 

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