TSTST 2013 Problem 2

A finite sequence of integers $a_{1}, a_{2}, \ldots, a_{n}$ is called regular if there exists a real number $x$ satisfying $$\lfloor k x\rfloor=a_{k} \text { for } 1 \leq k \leq n$$ Given a regular sequence $a_{1}, a_{2}, \ldots, a_{n}$, for $1 \leq k \leq n$ we say that the term $a_{k}$ is forced if the following condition is satisfied: the sequence $$a_{1}, a_{2}, \ldots, a_{k-1}, b$$ is regular if and only if $b=a_{k}$. Find the maximum possible number of forced terms in a regular sequence with 1000 terms.