Generalized specialized q,t-rectangle formula

For $a,b\geq 0$, write $b=b'k$ and $a=a'k$ for integers $a',b',k\geq 0$ with $\gcd(a',b')=1$, and let $S_{a,b}(q,t)$ be the q,t-rectangle number. Let $[j]_{q^k}$ denote the q-integer with base $q^k$, and let $\genfrac{[}{]}{0pt}{}{a+b}{a,b}_q$ be the q-multinomial coefficient. Generalized specialized q,t-rectangle formula.

$$q^{k(a'-1)(b'-1)/2+a'b'\binom{k}{2}}S_{a,b}(q,1/q)=\frac{a'+b'}{[a'+b']_{q^k}}\genfrac{[}{]}{0pt}{}{a+b}{a,b}_q.$$

This extends the preceding specialized formula from rectangles with dimensions $n$ and $mn$ to arbitrary rectangles, with the common-factor decomposition controlling the exponent and denominator.