
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  $n$  denote the product of the first  $2013$  primes.  Find the sum of all primes  $p$  with  $20 \le p \le 150$  such that
(i)  $\frac{p+1}{2}$  is even but is not a power of  $2$ , and 
(ii) there exist pairwise distinct positive integers  $a,b,c$  for which \[ a^n(a-b)(a-c) + b^n(b-c)(b-a) + c^n(c-a)(c-b) \] is divisible by  $p$  but not  $p^2$ .

*Proposed by Evan Chen*

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