Searching for multiple words in a cell

Hello,

I'm trying to search for a list of words within a cell that contains text and have those words displayed in the column adjacent to it.

For example:

A1: The cat in the hat.
A2: The dog in the hat.
A3: The walrus in the hat.
A4: The cat on the mat.
A5: The cat and dog on the mat.

B1:B5 contains the search terms I'm looking for:
B1: Cat
B2: Dog
B3: Walrus
B4: Hat
B5: Mat

I would like C1:C5 to show whether the search terms in B1:B5 are found in A1:A5, and then display those results (ie C1 should show 'cat and hat' and C2 should show 'dog and hat' and C5 should show 'cat, dog and mat').

Currently I'm using =IF(COUNT(SEARCH({"Cat","Dog"},A1)),"Cat","") but this formula isn't particularly useful for my purposes since it is TRUE/FALSE rather than returning the actual search term.

I would be very grateful for any help - thank you so much in advance!