How to use the pattern に行く to express several purposes


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.




> 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.)




> 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.)

