Solve the system of linear inequalities with parameters

$\begin{array}{}\text{(*)}& \{\begin{array}{l}0\le \phantom{-2\phantom{\rule{thickmathspace}{0ex}}}x+2\phantom{\rule{thinmathspace}{0ex}}y-3\phantom{\rule{thinmathspace}{0ex}}b+3\phantom{\rule{thinmathspace}{0ex}}a\le 2\\ 0\le -2\phantom{\rule{thinmathspace}{0ex}}x-3\phantom{\rule{thinmathspace}{0ex}}y+6\phantom{\rule{thinmathspace}{0ex}}b\phantom{\phantom{\rule{thickmathspace}{0ex}}+3a\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}}\le 1\\ 0\le x\le 1\\ 0\le y\le 2\\ 0\le a\le 1\\ 0\le b\le 1\end{array}\end{array}$

Here x,y are unknown variables and a,b are parameters.

My attempt. By adding the inequalities with some coeficients I separated the variables and get the simple system

$\begin{array}{}\text{(**)}& \{\begin{array}{l}0\le y+6a\le 5,\\ 0\le -x+9a+3b\le 8.\end{array}\end{array}$

and I am able to solve it. But the solutions of the last system are not solution of the initial system!

Maple and wolframAlpha cant solve the system.

P.S.1 For $a=\frac{63}{100}$ and $b=\frac{59}{100}$ Maple gives the solutions

$\begin{array}{c}\{x=1,\frac{9}{50}\le y,y\le \frac{11}{25}\},\{x=-3/2\phantom{\rule{thinmathspace}{0ex}}y+\frac{127}{100},\frac{9}{50}<y,y<\frac{11}{25}\},\{\frac{9}{50}<y,x<1,y<\frac{11}{25},-3/2\phantom{\rule{thinmathspace}{0ex}}y+\frac{127}{100}<x\},\{y=\frac{11}{25},\frac{61}{100}\le x,x<1\},\{x=-3/2\phantom{\rule{thinmathspace}{0ex}}y+\frac{127}{100},\frac{11}{25}<y,y<\frac{127}{150}\},\{\frac{11}{25}<y,x<-2\phantom{\rule{thinmathspace}{0ex}}y+\frac{47}{25},y<\frac{127}{150},-3/2\phantom{\rule{thinmathspace}{0ex}}y+\frac{127}{100}<x\},\{x=-2\phantom{\rule{thinmathspace}{0ex}}y+\frac{47}{25},\frac{11}{25}<y,y<\frac{127}{150}\},\{x=0,\frac{127}{150}\le y,y\le \frac{47}{50}\},\{y=\frac{127}{150},x\le \frac{14}{75},0<x\},\{0<x,\frac{127}{150}<y,x<-2\phantom{\rule{thinmathspace}{0ex}}y+\frac{47}{25},y<\frac{47}{50}\},\\ \{x=-2\phantom{\rule{thinmathspace}{0ex}}y+\frac{47}{25},\frac{127}{150}<y,y<\frac{47}{50}\}\end{array}$