Find all real values of the parameter $a$ for which the system \begin{align*} &1+\left(4x^2-12x+9\right)^2+2^{y+2}=a\\ &\log_3\left(x^2-3x+\frac{117}4\right)+32=a+\log_3(2y+3) \end{align*}has a unique real solution. Solve the system for these values of $a$.