814. 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 $998244353$.

814. 中强音

$4n$ 人低着头等距站在某圆上。当钟声响起时，所有人瞬间抬头，并且立刻看向最靠近他的两人、正对着他的一人这三人其中之一。如果二人对视，那么这两人会立刻尖叫。

记 $S(n)$ 为正好有 $2n$ 人尖叫的方法数。已知 $S(1) = 48$ 且 $S(10) \equiv 420121075 \mod{998244353}$。

求 $S(10^3)$ 模 $998244353$ 之值。

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