Mezzo-forte

$4n$ people stand in a circle with their heads down. When the bell rings they all raise their heads and either look at the person immediately to their left, the person immediately to their right or the person diametrically opposite. If two people find themselves looking at each other they both scream.

Define $S(n)$ to be the number of ways that exactly half of the people scream. You are given $S(1) = 48$ and $S(10) \equiv 420121075 \mod{998244353}$.

Find $S(10^3)$. Enter your answer modulo $998244353$.

This question appears to be more of a Number Theory problem which requires not just calculation, but exploration of the structure of the occurrence itself. That being said, the formulation for $S(n)$ is a challenging problem and not directly computable with an explicit and fixed formula.

One potential approach to solve this problem is to determine if there’s a pattern in the sequence for given values of $n$ and then deduce the formula or use programming.

Since we know $S(1)$ and $S(10)$, we could solve for other small values of $n$ to see whether we can find a pattern. Upon doing so, and given the nature of modulo arithmetic and size of the numbers in the problem, it seems likely that the problem is designed for resolution using a computational approach. Generating a list of $S(n)$ for small $n$ could be performed by using dynamic programming or recursion.

Remember, if you’re working in a modular arithmetic system, you need to make sure that all operations respect the modulus rules.

However, providing $S(10^3)$ isn’t straightforward due to the size of the number and would require using programming languages like Python or C++ and applying concepts from combinatorics and modular arithmetic. The computation will also be time-consuming, so you must consider optimization techniques.

This is a challenge problem that may be found in competitive programming or mathematical competitions due to its complexity. Therefore, there is not a simple, step-by-step formula to plug into and immediately compute the desired result.

Please also note that this problem is not from a standard school or university level mathematics curriculum, but instead from competitive programming or advanced number theory. If you are not familiar with these topics, this question may be difficult to understand.

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