godan verb (-su), intransitive verb
go out with a purpose; set out to do
Compound of 行く (to go) and 出す (to start doing). Used when someone goes somewhere specifically to do something, often implying a deliberate departure.
彼は買い物に行き出した。
He went out to do some shopping.
雨がやんだので、散歩に行き出した。
Since the rain stopped, I set out for a walk.