What is the smallest $\gamma_1,\gamma_2,\gamma_3>0$ such that given coprime $p,q=\Theta(\ell)$ and integer $t\geq3$ there are coprime $m,n=\Theta(\ell^{t-1})$ with $(mn,pq)=1$, $\alpha_i\in\Bbb Z$ and $m'\in\Bbb Z$ with
$$\alpha_1p^{t-2}q^t+\gamma_1=m'm^2$$
$$\alpha_2p^{t-1}q^{t-1}+\gamma_2=m'mn$$
$$\alpha_3p^tq^{t-2}+\gamma_3=m'n^2$$
where $m,n$ where $0<m'<p^{t}q^t$ and $\alpha_i=O(\ell^{2t})$ holds?