Expectation of $\frac{X_1}{\sqrt{X_1^2+\cdots+X_n^2}}$ is $0$ if $E(X_1)=0$ and random variables are i.i.d

Question

Let $X_1\cdots X_n$ be independent identically distributed random variables and $E(X_1)=0$ and $Var(X_1)<\infty$. Do we have that $E(\frac{X_1}{\sqrt{X_1^2+\cdots+X_n^2}})=0$?

My attempt: $E(\frac{X_1}{\sqrt{X_1^2+\cdots+X_n^2}})=\sum_m E(\frac{X_1}{m}|\sqrt{X_1^2+\cdots+X_n^2}=m)P(\sqrt{X_1^2+\cdots+X_n^2}=m)=\sum_m\frac{1}{m} E(X_1|\sqrt{X_1^2+\cdots+X_n^2}=m)P(\sqrt{X_1^2+\cdots+X_n^2}=m)$

So if $E(X_1|\sqrt{X_1^2+\cdots+X_n^2}=m)=0$ then we're done. But I am not sure if this is correct or not.

Thanks for any hint.