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Let \( a_{1}, a_{2}, \ldots, a_{n} \) be real constants, \( x \) be a real variable and \( f(x)=\cos \left(a_{1}+x\right)+\frac{1}{2} \cos \left(a_{2}+x\right)+\frac{1}{4} \cos \left(a_{3}+x\right)+ \)
\[
\ldots+\frac{1}{2^{n-1}} \cos \left(a_{n}+x\right)
\]
Given that \( f\left(x_{1}\right)=f\left(x_{2}\right)=0 \), prove that \( x_{2}-x_{1}=m \pi \) for some integer \( m \).
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