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Is the number $$(11!)!+11!+1$$ a prime number ?

I do not expect that a probable-prime-test is feasible, but if someone actually wants to let it run, this would of course be very nice. The main hope is to find a factor to show that the number is not prime. If we do not find a factor, it will be difficult to check the number for primality. I highly expect a probable-prime-test to reveal that the number is composite. "Composite" would be a surely correct result. Only if the result would be "probable prime", there would remain slight doubts, but I would be confident with such a test anyway.

Motivation : $(n!)!+n!+1$ can only be prime if $\ n!+1\ $ is prime. This is because a non-trivial factor of $\ n!+1\ $ would also divide $\ (n!)!+n!+1\ $. The cases $\ n=2,3\ $ are easy , but the case $\ n=11\ $ is the first non-trivial case. We only know that there is no factor upto $\ p=11!+1\ $

What I want to know : Can we calculate $$(11!)!\mod \ p$$ for $\ p\ $ having $\ 8-12\ $ digits with a trick ? I ask because pari/gp takes relatively long to calculate this residue directly. So, I am looking for an acceleration of this trial division.