Lattices and Farey Sequences

[USA TSTST 2013/2] A finite sequence of integers $a_1, a_2, \dots, a_n$ is called regular if there exists a real number $x$ satisfying $$\left\lfloor kx \right\rfloor = a_k \quad \text{for } 1 \le k \le n.$$ Given a regular sequence $a_1, a_2, \dots, a_n$, for $1 \le k \le n$ we say that the term $a_k$ is forced if the following condition is satisfied: the sequence $$a_1, a_2, \dots, a_{k-1}, b$$ is regular if and only if $b = a_k$. Find the maximum possible number of forced terms in a regular sequence with $1000$ terms.