The sequence \(a_1,a_2,\dots\) of integers satisfies the conditions:

(i) \(1\leq a_j\leq 2015\) for all \(j\geq1\), (ii) \(k+a_k\not = \ell+ a_\ell\) for all \(1\leq k< \ell\).

Prove that there exist two positive integers \(b\) and \(N\) for which

\[\left \vert \sum_{j=m+1}^n (a_j-b) \right\vert \leq1007^2\]

for all integers \(m\) and \(n\) such that \(n>m\geq N\).

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