Heron is going to watch a show with $n$ episodes which are released one each day. Heron wants to watch the first and last episodes on the days they first air, and he doesn’t want to have two days in a row that he watches no episodes. He can watch as many episodes as he wants in a day. Denote by $f(n)$ the number of ways Heron can choose how many episodes he watches each day satisfying these constraints. Let $N$ be the $2021$st smallest value of $n$ where $f(n) \equiv 2$ mod $3$. Find $N$.

Let's think step by step and output the final answer within \boxed{}.
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