Exciting Game Announcement!

We are thrilled to introduce our latest game, which promises to bring hours of fun and entertainment. This game is available exclusively for PC.

To gain access to the game, you must first solve the following problem:

Problem: Let $P (x)$ be a polynomial of degree 2024 defined by:

$P (x) = x^{2024} + a_{2023} x^{2023} + a_{2022} x^{2022} + \dots + a_{1} x + a_{0},$

where all coefficients $a_{i}$ are integers. It is known that for any integer $n$ , $P (n)$ is divisible by $n^{2} + n + 1$ . Determine the sum of all the coefficients of the polynomial $P (x)$ such that $- a_{2023} < 3^{997}$ .

Important Notice: This game is intended solely for entertainment and should not be used during work hours. Please play responsibly and enjoy your time!