seek LOG 0,0;
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will reset your LOG filehandle to the top, but you may need to adjust your nested if loops. Some pseudocode(untested)

while ( <LOG> ) {
 my $LINE = $_;
  if ( $LINE =~ m/002389983/ ) {
   print "Looking for: 002389983\n Found it in file: $LINE\n";
   last;
   }
}

seek LOG, 0, 0;

while ( <LOG> ) {
  if ( $LINE =~ m/29994339499/ ) {
    print "Looking for: 29994339499\n Found it in file: $LINE\n";
    }else {
     print "Could not find the second string: 29994339499\n";
   }
  }
[download]