Proving a duality between Ext and Tor for maximal Cohen-Macaulay modules over Gorenstein ring

Let $(R,mathfrak m, k)$ be a local complete Gorenstein ring of dimension $d$. Let $M,N$ are maximal Cohen-Macaulay modules (i.e. have depth equal to $d$) that are locally free on the punctured spectrum (i.e. $M_P, N_P$ are free over $R_P$ for every non-maximal prime ideal $P$ of $R$) . Let $E(k)$ be the injective hull of the residue field $k$.

Then, how to prove that

$$text{Ext}^d_R( text{Tor}_i^R(M,N^*), R)cong text{Ext}^{d+i}_R(M,N),forall ige 1$$ ?

Here $(-)^*:=text{Hom}(-, R)$

My thoughts: Let us write $(-)^{lor}:=text{Hom}(-,E(k))$.
Since $M,N$ are locally free on the punctured spectrum, so $text{Tor}_i^R(M,N^*)$ has finite length for every $i>0$. So $H^0_{mathfrak m}(text{Tor}_i^R(M,N^*))cong text{Tor}_i^R(M,N^*)$ . So by local duality, we get

$text{Ext}^d_R( text{Tor}_i^R(M,N^*), R)cong (H^0_{mathfrak m}(text{Tor}_i^R(M,N^*))^{lor} cong ( text{Tor}_i^R(M,N^*))^{lor}cong text{Tor}_i^R(M, H^d_{mathfrak m}(N)^{lor})^{lor} cong text{Ext}^i_R(M, H^d_{mathfrak m}(N))$

So basically we’re trying to prove $text{Ext}^i_R(M, H^d_{mathfrak m}(N))cong
text {Ext}^{d+i}_R(M,N),forall ige 1$.

Also note that for any module $M$, we have a stable isomorphism $syz^2 text{Tr} M cong M^*$ , where $syz^2(-)$ denotes second syzygy and $text{Tr }(-)$ denotes Auslander transpose. So,
$text{Tor}_i^R(M,N^*)cong text{Tor}_{i+2}^R(M, text {Tr }N)$ .

But I’m unable to simplify things further.

One key point that might be useful is that over Gorenstein local rings, maximal Cohen-Macaulay modules are reflexive and their duals are again maximal Cohen-Macaulay.

Please help.