Which impact on security (factorization) has a common prime factor among the prime factors $P$,$Q$ of a number $N$ $$N=P\cdot Q$$ $$P=2\cdot F\cdot p+1$$ $$Q=2\cdot F\cdot q+1$$ with $F,q,p$ different primes and $F$ the biggest prime factor of $P$ and $Q$ with $$F\gg p,q$$
A potential adversary who want to factorize $N$ does know about the internal structure but does not know $F,p,q,P,Q$
For example $N$ is a $1024$-bit number with $P,Q \approx$ $512$-bit each.
$F \approx 461$-bit and $p,q \approx 50$-bit each.
Would security significantly change for larger $N,F$ but constant size $p,q$?
Or how would security change for larger/smaller $p,q$ but constant size of $N$?
Edit Update: It turned out a common factor is not necessary. I did some more detailed question.