Congruences modulo non-Chen primes

Question

A Chen prime $s$ is a prime such that $s+2$ is either prime or semi-prime. A non-Chen prime is a prime such that $s+2$ is not a prime or a semi-prime. The prime $733$ is congruent to $1$ mod 61, to $2$ mod $43$ and to $3$ mod $73$. $43, 61, 73$ are all consecutive non-Chen primes. First of all, I wonder if for all primes p<$733$, p is congruent to at most $1,2$ modulo two consecutive non-Chen primes. And I wonder if there is a prime congruent to $1,2,3,4$ modulo four consecutive non-Chen primes. Is the Chinese-remainder theorem useful?