# Leet Sum

Algebra Level 5

The sequence $$\{a_k\}_{k=1}^{112}$$ satisfies $$a_1 = 1$$ and $$a_n = \dfrac{1337 + n}{a_{n-1}}$$, for all positive integers $$n$$. Let

$S = \left\lfloor a_{10}a_{13} + a_{11}a_{14} + a_{12}a_{15} + \cdots + a_{109}a_{112}\right\rfloor.$

Find the remainder when $$S$$ is divided by $$1000$$.

This problem is posed by Akshaj.

Details and assumptions

The function $$\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}$$ refers to the greatest integer smaller than or equal to $$x$$. For example $$\lfloor 2.3 \rfloor = 2$$ and $$\lfloor -5 \rfloor = -5$$.

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