what is the difference between 'find', 'discover' and 'find out'

English as a Second Language English Vocabulary, Grammar and Idioms
1

I saw that they had discover a new route.
I saw that they had found a new route.
I saw that they had found out a new route.

2

(1) should be ‘discovered’ as it is in the past.

‘Discovered’ and ‘found’ are synonyms there. ‘Found out’ is incorrect as this phrasal verb does not carry that sense.