Let \(Q\) be a set of prime numbers, not necessarily finite. For a positive integer \(n\) consider its prime factorization: define \(p(n)\) to be the sum of all the exponents and \(q(n)\) to be the sum of the exponents corresponding only to primes in \(Q\). A positive integer \(n\) is called special if \(p(n)+p(n+1)\) and \(q(n)+q(n+1)\) are both even integers. Prove that there is a constant \(c>0\) independent of the set \(Q\) such that for any positive integer \(N>100\), the number of special integers in \([1,N]\) is at least \(cN\).

(For example, if \(Q=\{3,7\}\), then \(p(42)=3\), \(q(42)=2\), \(p(63)=3\), \(q(63)=3\), \(p(2022)=3\), \(q(2022)=1\).)