3. Given the function $f(x)=\sin \omega x+\sin 2 x$, where $\omega \in \mathbf{N}_{+}, \omega \leqslant 2023$. If $f(x)<2$ always holds, then the number of constants $\omega$ that satisfy the condition is $\qquad$
When $\sin 2 x=1$, $2 x=2 k \pi+\frac{\pi}{2} \Rightarrow x=k \pi+\frac{\pi}{4}$, at this time $\omega x=\omega k \pi+\frac{\omega \pi}{4} \neq 2 m \pi+\frac{\pi}{2}$. When $(4 k+1) \omega=8 m+2$, let $\omega=2 t \Rightarrow t(4 k+1)=4 m+1$, then $t$ is an odd number that is 1 modulo 4, so $\omega$ is an even number that is 2 modulo 8. Conversely, this is obviously true.

Therefore, the number of constants $\omega$ that satisfy the condition is $2023-253=1770$.