"Your task is to find the exit."
Translation:Waszym zadaniem jest znaleźć wyjście.
I would argue that while it shouldn't be the default translation it should be accepted – Meaning of the is pretty close to the role "to" has in "Twoim zadaniem jest znaleźć to wyjście" in that both specify some concrete exit that must be found, not just any exit. While normally translating the as "to" is an overkill, there are context where it is justified, like translating "The Elvis" as "Ten Elvis" to mean Elvis Presley.
Native here: It does not work, it's broken. The best transcription into English I can come up with is "Your task is find the exit." In this particular sentence structure I cannot come up with any single verb that would be grammaticaly correct. You have to use a noun in the secondary clause.
EDIT: After a break I remembered about a class of verbs that do work. It is when the task or duty given are perpetual (imperfective?) actions. But when I think of those in terms of grammar, I believe they are also treated as nouns, with the same spelling as their counterparty verbs. "ochrona" "budowa" "nauka" and so on.
For this construction, there are only several results in the corpus (http://www.nkjp.pl/poliqarp/nkjp300/query/ - search for zadanie to [pos=inf]), but I think they show it's valid, although rare and probably not something I'd advice to use.
By 'several' I mean 4-5, because some of 10 results shown use the word 'to' in a different meaning.
Znaleźć is a verb, so it can't take any case. It can replace the verbal noun znalezienie.
The rule is:
If you use jest/są instead of to: [Nominative noun] is a subset of [instrumental noun].
That is why:
Twoim zadaniem jest znalezienie wyjścia.
In this sentence, finding the exit is a subcategory of task and not the other way round; this is why you can't switch the cases.