linux rows by column statistics

I have a document , the contents of the following form:
aa bb cc dd ee
aa 34 cc 12 34
ee 34 44 55 66
.........
Now I have to follow the first and third columns statistics ( first and third columns if the same , leaving only one line ) , output of the first row and the first four , and the number of records , such as the above output :
aa cc 2
ee 44 1

beg you big God
------ Solution -------------------------------- ------------
accordance with your request output , where test is the documentation of your input data

$ cat test |awk '{print $1" "$3}' | uniq -c | awk '{print $2" "$3" "$1}'

aa cc 2
ee 44 1

------ For reference only ---------------------------------- -----

$ cat test

aa bb cc dd ee
aa 34 cc 12 34
ee 34 44 55 66

$ cat test |awk '{print $1" "$3}' | uniq -c

2 aa cc
1 ee 44

number on the front
------ For reference only ---------------------------- -----------
thanks ah