How can I make a certain 2D plot?

I want to draw this elementary graph, shown below.

The exact path of the blue line is not important. I want to keep the blue dots to the left of the line and keep the red dots to the right of it.

I have tried the following code, but it failes to properly curve the straight line to keep all the blue dots to the left.

k = 3;
poles1 = Table[{i, 0}, {i, 0, k}];
poles2 = Table[{1/2 - n, 0}, {n, 0, k}]; 
Show[
  Plot[100 Sign[x + 1/4], {x, -k, k}, 
    ExclusionsStyle -> Blue, 
    PlotRange -> {-3, 3}, 
    AxesStyle -> Directive[Blue, 13]], 
  Graphics[{PointSize[0.02], Red, Point[poles1]}], 
  Graphics[{PointSize[0.02], Blue, Point[poles2]}]]

Any help will be highly appreciated.

k = 3;
poles1 = Table[{i, 0}, {i, 0, k}];
poles2 = Table[{1/2 - n, 0}, {n, 0, k}]; 
Show[Graphics[{PointSize[0.02], Red, Point[poles1]}], 
  Graphics[{PointSize[0.02], Blue, Point[poles2]}], Axes -> True, PlotRange -> 3, 
  AxesStyle -> Directive[Blue, 13]] // Canvas

With[{k = 3},
  poles1 = Table[{i, 0}, {i, 0, k}]; 
  poles2 = Table[{1/2 - n, 0}, {n, 0, k}]];

Graphics[
  {{PointSize[0.02],
     {Red, Point[poles1]}, {Blue, Point[poles2]}},
   {Blue,
     HalfLine[{-1/4, 0}, {0, -1}],
     Circle[{0, 0}, 1/4, {0, π}], Circle[{1/2, 0}, 1/4, {π, 2 π}], 
     Circle[{0, 0}, 3/4, {0, π/2}], Circle[{0, 1}, 1/4, {-π/2, -π}], 
     HalfLine[{-1/4, 1}, {0, 1}]}},
  Axes -> True,
  PlotRange -> 3]