if $\sin{(aw)}+\sin{(bw)}+\sin{(cw)}=3$ find $w$ range let $w$ is postive integer,if there exist $a,b,c(\pi\le a<b<c\le 2\pi)$such
$$\sin{(aw)}+\sin{(bw)}+\sin{(cw)}=3$$
Find the $w$ range.
My attempt：
since
$$\sin{(wa)}\le 1,\sin{(wb)}\le 1,\sin{(wc)}\le 1$$
then $$\sin{(wa)}=\sin{(wb)}=\sin{(wc)}=1$$
$$aw=\dfrac{\pi}{2}+2k_{1}\pi,k_{1}\in Z$$
$$bw=\dfrac{\pi}{2}+2k_{2}\pi,k_{2}\in Z$$
$$cw=\dfrac{\pi}{2}+2k_{3}\pi,k_{3}\in Z$$
what approaches do you think, I could take to solving the next step?

We have
$$\begin{align}&\sin(aw)+\sin(bw)+\sin(cw)=3\\\\&\iff \sin(aw)=\sin(bw)=\sin(cw)=1\\\\&\small\iff\text{There exist three integers $k,l,m$ such that $aw=\frac{\pi}{2}+2k\pi,bw=\frac{\pi}{2}+2l\pi,cw=\frac{\pi}{2}+2m\pi$}\\\\&\small\iff\text{There exist three integers $k,l,m$ such that $\pi\le\frac{\pi+4k\pi}{2w}\lt\frac{\pi+4l\pi}{2w}\lt\frac{\pi+4m\pi}{2w}\le 2\pi$}\\\\&\small\iff \text{There exist three integers $k,l,m$ such that $2\le\frac{1+4k}{w}\lt\frac{1+4l}{w}\lt\frac{1+4m}{w}\le 4$}\end{align}$$
So, let us find a condition for $w$ such that there exist at least three distinct positive integers $n$ satisfying $2\le\frac{1+4n}{w}\le 4$, i.e. $\frac{2w-1}{4}\le n\le \frac{4w-1}{4}$.
For $w=2N$ where $N$ is a positive integer, the condition is
$$\left\lfloor\frac{4w-1}{4}\right\rfloor-\left\lceil\frac{2w-1}{4}\right\rceil+1\ge 3\iff (2N-1)-N+1\ge 3\iff N\ge 3.$$
For $w=2N-1$ where $N$ is a positive integer, the condition is
$$\left\lfloor\frac{4w-1}{4}\right\rfloor-\left\lceil\frac{2w-1}{4}\right\rceil+1\ge 3\iff (2N-2)-N+1\ge 3\iff N\ge 4.$$
Hence, the answer is $\color{red}{w\ge 6}$.
