You are an expert Verilog hardware designer working on a complete FPGA or ASIC project. You are given the full source code of a Verilog project. One of the modules in the project is incomplete — its implementation is missing and marked with:

// >>> Module Implementation Begin

// <<< Module Implementation End

Your task is to complete the missing implementation so that the entire project functions correctly. You must infer what the missing module or logic is supposed to do by analyzing the rest of the codebase: module names, signal connections, instantiations, comments, and usage patterns.

Follow these rules:

1. Only generate Verilog code to go between the `// >>> Module Implementation Begin` and `// <<< Module Implementation End` comments.
2. Use Verilog-2001 syntax. Ensure the code is synthesizable.
3. Assume the goal is a working, integrated hardware design — your code must make the whole system function as intended.
4. Analyze the naming, I/O ports, and how the module is instantiated or used in other parts of the project.
5. Implement correct behavior based on design context. Do not guess randomly — reason it out.
6. Avoid vendor-specific features unless explicitly required.
7. Do not output comments, explanation, or anything outside the required code region unless explicitly asked.

Your output should contain **only the Verilog code** that fits inside the marked region.


module BigCircle(output G, P, input Gi, Pi, GiPrev, PiPrev);
// >>> Module Implementation Begin

// <<< Module Implementation End
endmodule



module SmallCircle(output Ci, input Gi);
  buf (Ci, Gi);
endmodule

module Square(output G, P, input Ai, Bi);
  and (G, Ai, Bi);
  xor (P, Ai, Bi);
endmodule

module Triangle(output Si, input Pi, CiPrev);
  xor (Si, Pi, CiPrev);
endmodule


module tt_um_prefix8 (
    input  wire [7:0] ui_in,    // Dedicated inputs
    output wire [7:0] uo_out,   // Dedicated outputs
    input  wire [7:0] uio_in,   // IOs: Input path
    output wire [7:0] uio_out,  // IOs: Output path
    output wire [7:0] uio_oe,   // IOs: Enable path (active high: 0=input, 1=output)
    input  wire       ena,      // always 1 when the design is powered, so you can ignore it
    input  wire       clk,      // clock
    input  wire       rst_n     // reset_n - low to reset
);
wire [7:0] sum, a, b;
wire cout;
wire cin = 1'b0;
wire [7:0] c;
wire [7:0] g, p;

assign a = ui_in;
assign b = uio_in;

Square sq[7:0](g, p, a, b);

wire [15:8] g1, p1;
BigCircle bc1_8(g1[8], p1[8], g[1], p[1], g[0], p[0]);
BigCircle bc1_11(g1[11], p1[11], g[5], p[5], g[4], p[4]);
BigCircle bc1_13(g1[13], p1[13], g[7], p[7], g[6], p[6]);
BigCircle bc1_15(g1[15], p1[15], g[4], p[4], g[3], p[3]);

wire [9:9] g2, p2;
BigCircle bc2_9(g2[9], p2[9], g[2], p[2], g1[8], p1[8]);

wire [16:10] g3, p3;
BigCircle bc3_10(g3[10], p3[10], g[3], p[3], g2[9], p2[9]);
BigCircle bc3_16(g3[16], p3[16], g1[15], p1[15], g2[9], p2[9]);

wire [12:12] g4, p4;
BigCircle bc4_12(g4[12], p4[12], g1[11], p1[11], g3[10], p3[10]);

wire [17:14] g5, p5;
BigCircle bc5_14(g5[14], p5[14], g1[13], p1[13], g4[12], p4[12]);
BigCircle bc5_17(g5[17], p5[17], g[6], p[6], g4[12], p4[12]);

SmallCircle sc0(c[0], g[0]);
SmallCircle sc1(c[1], g1[8]);
SmallCircle sc2(c[2], g2[9]);
SmallCircle sc3(c[3], g3[10]);
SmallCircle sc4(c[4], g3[16]);
SmallCircle sc5(c[5], g4[12]);
SmallCircle sc6(c[6], g5[17]);
SmallCircle sc7(c[7], g5[14]);
Triangle tr0(sum[0], p[0], cin);
Triangle tr1(sum[1], p[1], c[0]);
Triangle tr2(sum[2], p[2], c[1]);
Triangle tr3(sum[3], p[3], c[2]);
Triangle tr4(sum[4], p[4], c[3]);
Triangle tr5(sum[5], p[5], c[4]);
Triangle tr6(sum[6], p[6], c[5]);
Triangle tr7(sum[7], p[7], c[6]);

buf (cout, c[7]);

assign uo_out = sum;
assign uio_out = 8'b00000000;
assign uio_oe = 8'b00000000;
endmodule

Reference solution:
rtl/generated.v:   wire e;
  and (e, Pi, GiPrev);
  or (G, e, Gi);
  and (P, Pi, PiPrev);