Retrieve message from rsa ciphertext based on some conditions Assume:

$x = m^{e-1}\bmod n$
$y = m^{d-1}\bmod n$

Here $m$ is a message, $e$ is the public exponent, $d$ is the private key and $n$ is the modulus of an RSA key pair.
Now if I know $e=65537$, $x$, $y$, and $n$, can I retrieve the message $m$?
let \begin{equation}
x*y=m^{e+d-2} \mod n \qquad (1)
\end{equation}
\begin{equation}
m(x+y)=(m^{e}+m^{d}) \mod n \qquad (2)
\end{equation}taking $log_m $ for each from (2) side with respect to $\mod n$, we will have
\begin{equation}
log_mm(x+y)=log_m(m^{e}+m^{d}) \mod n \qquad 
\end{equation}
\begin{equation}
1+log_m(x+y)=log_m(m^{e})+log_m(1+\frac{m^d} {m^e}) \mod n \qquad 
\end{equation}
\begin{equation}
1+log_m(x+y)\ge log_m(m^{e})+log_m(\frac{m^d} {m^e}) \mod n \qquad 
\end{equation}
\begin{equation}
1+log_m(x+y)\ge e+d-e \mod n \qquad 
\end{equation}
\begin{equation}
d\simeq  1+log_m(x+y)\qquad (3)
\end{equation}
substitute $(3)$ in $(1)$
\begin{equation}
x*y=m^{e+(1+log_m(x+y))-2} \mod n \qquad 
\end{equation}
\begin{equation}
x*y=m^{e-1}m^{log_m(x+y)} \mod n \qquad 
\end{equation}
\begin{equation}
m^{e-1}=(x+y)^{-1}(x*y) \mod n \qquad 
\end{equation}