
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 prime  $p$ , a natural number  $n$  and an integer  $a$ , we let  $S_n(a,p)$  denote the exponent of  $p$  in the prime factorisation of  $a^{p^n} - 1$ .  For example,  $S_1(4,3) = 2$  and  $S_2(6,2) = 0$ .  Find all pairs  $(n,p)$  such that  $S_n(2013,p) = 100$ .

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