Obtaining multivariate linear regression coefficients from residual values of simple linear regressions

Question

We are interested in a multivariate linear regression of $Y$ against $X_1$ and $X_2$. More specifically we want to know the regression coefficients of $X_1$. There is a sample drawn of size $n$, but we don't have $n$ observations of $X_1$ and $X_2$, instead we are given the following for each observation in the sample:

• The residual value of $X_1$, after having regressed $X_1$ against $X_2$
• The residual value of $X_2$, after having regressed $X_2$ against $X_1$
• $Y$

How do you proceed?

Thanks in advance.