How to use the pattern に行く to express several purposes

Question

I was reviewing the stem + に行く like for example:

うちへ食べに行く.

But, how about expressing several purposes like:

I'm going home to eat and take a shower.

I'm not really sure how to add several purposes with this pattern, or if its even possible.

Is the next example even grammatically correct?

うちへ食べとシャワーを浴びに行く。

Thank you beforehand.

Answer 1

うちに食べに帰る。/ ｛ごはん/お昼etc｝を食べにうちに帰る。
I'm going home to eat (supper, lunch etc).
うちにシャワーを浴びに帰る。/ シャワーを浴びにうちに帰る。
I'm going home to take a shower.

sound good, but...

うちに食べとシャワーを浴びに帰る。
うちに食べてシャワーを浴びに帰る。
食べてシャワーを浴びにうちに帰る。*

sound incorrect/unnatural.
_{(*食べてシャワーを浴びにうちに帰る would be understood as "I will eat (something/supper) and go home to take a shower.")}

I'm going home to eat and take a shower.

I can't think of a way to say it using 「連用形+に+行く/帰る」. I would instead say...

うちに帰って、｛ごはん/お昼/お昼ご飯/晩ご飯etc｝を食べてシャワーを浴びる/浴びてくる。*

_{(*You'd use ～てくる when you're going to come back to where you are after you eat lunch and take a shower.)}