Does the solution to the following system of inequalities **exist**?

$$a-1\geq a\left( b_ic -d_i\right)\geq 1$$

where $a\in \mathbb{N}_{\geq3}$, $c\in \mathbb{R}$ and $b_i,d_i \in \mathbb{N}$. Moreover, $0< c < 2\pi$ and $0\leq d_i < b_i$ and $i$ runs from $1$ to $a-1$ because of which we have a system of $a-1$ inequalities. The unknowns are $c$ and $d_i$.

**Update:** With some consideration, the bounds can be improved. Instead of $0 < c < 2\pi$ and $0 \leq d_i < b_i$, it is sufficient enough to consider $0 < c \leq \frac{1}{2}$ and $0 \\leq d_i < \lfloor \frac{b_i-1}{2} \rfloor$.