Countifs formulas - use a range of values as the criteria

I'm trying to use a range of text values as the criteria in a countifs formula but haven't been successful. There are multiple ranges and criteria being evaluated but this is the only piece i can't get to work.
Example
A1 = Dog
A2 = Cat
A3 = Rabbit
A4 = Horse

basic formula i tried using: countifs(range1,criteria1,K:K,A1:A4,range3,criteria3,etc)
I'm not sure if this is helpful (as actually posting in a forum is completely new to me) but here is the whole actual formula with the problem area underlined:

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I found a similar post but the cells contained numerical values and the solutions seemed specific to that.
What i tried obviously doesn't work, but i'm curious if there's a way to achieve the same outcome. Basically, if row K contained any of the values in A1:A4, and met the other criteria, they would be counted.

That worked beautifully! I could hug you right now! Thanks so much!