changes the sign of a continuous $f(x)$ with period 2$\pi$ at least 2n+2 times on a interval $[a,b]$ with $b-a>2\pi$

Question

Let $f(x)$ is continuous on $\mathbb{R}$ with period $2\pi$, if there exists $n \geq 1, n \in \mathbb{N}$ we have \begin{equation} \int_0^{2\pi} f(x) dx = \int_0^{2\pi} f(x) \sin xdx= \int_0^{2\pi} f(x) \cos xdx=...= \int_0^{2\pi} f(x) \sin nx dx=\int_0^{2\pi} f(x) \cos nx dx=0 \end{equation}

for any interval $[a,b]$, for which $b-a > 2\pi$, prove that the changes of sign of the function $f(x)$ on $[a,b]$ at least $2n+2$ times