Exim is unable to route to any remote domains. It doesn't seen to be able to access the DNS.

Try running exim -d+resolver -bt ''remote address''. The -d options turns on debugging output, and the addition of +resolver will make it show the resolver queries it is building and the results of its DNS queries. If it appears unable to contact any name servers, check the contents and permissions of /etc/resolv.conf.

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