Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with their product; when it is Sara's turn, she can pick any two integers on the board and replace them with their sum. Play continues until exactly one integer remains on the board. Paul wins if that integer is odd, and Sara wins if it is even.

Initially, there are $2021$ integers on the board, each one sampled uniformly at random from the set $\{0, 1, 2, 3, \ldots, 2021\}$. Assuming both players play optimally, the probability that Paul wins is $\frac{m}{n}$, where $m, n$ are positive integers and $\gcd(m, n) = 1$. Find the remainder when $m + n$ is divided by $1000$.

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