Primes and factorials

Is there a prime number pp that satisfies this inequality?

2018!+2p2018!+20182018!+2 \le p \le 2018!+2018

Note: 2018!2018! is the factorial of 2018. That is, 2018!=1×2×3××2016×2017×2018.2018!=1 \times 2 \times 3 \times \cdots \times 2016 \times 2017 \times 2018.

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