this is my answer, I don't know if this can complete your answer or not:
(similar to falagar's answer )
Let wi be the product p2ip2i+1, where pi is ith prime, each wi is product of two
distinct primes and also wk and wt are coprime, now take t2 such that t2equiv−kpmodwifor k=−Ncdots0,1,2cdotsN,and i=1,2,cdots,2N+1 such that for k=−N,i=1 and,...,by chinese remainder theorem, we have such t2.
Now t2+i has at least two primes because t2+i is multiple of wi,since there are a prime between t and 2t and also t and t/2,so t+h1
and t−h2 are primes,so at least one of the numbers t2−N,cdotst2+1,t2+2cdots,t2+N is exactly product
of two distinct primes.
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