
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  $q$ , let  $\Phi_q(x)=x^{q-1}+x^{q-2}+\cdots+x+1$ . 
Find the sum of all primes  $p$  such that  $3 \le p \le 100$  and there exists an odd prime  $q$  and a positive integer  $N$  satisfying
\[\dbinom{N}{\Phi_q(p)}\equiv \dbinom{2\Phi_q(p)}{N} \not \equiv 0 \pmod p. \]*Proposed by Sammy Luo*

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