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.

/*
 * Copyright (c) 2024 Your Name
 * SPDX-License-Identifier: Apache-2.0
 */

`default_nettype none



module tt_um_multiplier_mbm (
    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
);
// >>> Module Implementation Begin

// <<< Module Implementation End
endmodule




module latchh(input [7:0]d,           // 1-bit input pin for data  
             input clk,          // 1-bit input pin for enabling the latch
              output reg [7:0]q);     // 1-bit output pin for data output  
  
   always @ (posedge clk)  
    q <= d;  
endmodule

module booth_encoding(input a,b,c,
                output zero,one,minus_one,two, minus_two);
wire a_n,b_n,c_n,a1,a2,a3,a4,a5,a6;
wire [7:0]i;

decoder_3_8 dec({a,b,c},i);

assign a_n=~a;
assign b_n=~b;
assign c_n=~c;

assign a1= a_n & b_n & c_n & i[0];
assign a2=a & b & c & i[7];
assign a3= a_n & b_n & c & i[1];
assign a4= a_n & b & c_n & i[2];

assign zero= a1 | a2;
assign one= a3 | a4;

assign a5= a & b_n & c & i[5];
assign a6= a & b & c_n & i[6];
assign minus_one= a5 | a6;

assign two= a_n & b & c & i[3];
assign minus_two=a & b_n & c_n & i[4];

endmodule

module decoder_3_8(input [2:0]din,
                    output reg[7:0]dout);
 always @( din )
        begin
          dout=8'd0;
          case (din)
              3'b000: dout[0]=1'b1;
              3'b001: dout[1]=1'b1;
              3'b010: dout[2]=1'b1;
              3'b011: dout[3]=1'b1;
              3'b100: dout[4]=1'b1;
              3'b101: dout[5]=1'b1;
              3'b110: dout[6]=1'b1;
              3'b111: dout[7]=1'b1;
              default: dout=8'd0;
          endcase
      end               
endmodule

module cla(input [15:0]x, input [13:0]y, input cin, output  [15:0]sum);
wire [13:0] p;
wire [13:0]g;
wire [13:1] c;

assign sum[0]= x[0];
assign sum[1]= x[1];

assign p[0]= x[2] ^ y[0];
assign g[0]= x[2] & y[0];
assign c[1]= g[0] | (p[0] & cin);
assign sum[2]= p[0] ^ cin;

assign p[1]= x[3] ^ y[1];
assign g[1]= x[3] & y[1];
assign c[2]= g[1] | (p[1] & g[0]) | (p[1] & p[0] & cin);
assign sum[3]= p[1] ^ c[1];

assign p[2]= x[4] ^ y[2];
assign g[2]= x[4] & y[2];
assign c[3]= g[2] | (p[2] & g[1]) | (p[2] & p[1] & g[0]) | (p[2] & p[1] & p[0] & cin);
assign sum[4]= p[2] ^ c[2]; 

assign p[3]= x[5] ^ y[3];
assign g[3]= x[5] & y[3];
assign c[4]= g[3] | (p[3] & g[2]) | (p[3] & p[2] & g[1]) | (p[3] & p[2] & p[1] & g[0]) | (p[3] & p[2] & p[1] & p[0] & cin);
assign sum[5]= p[3] ^ c[3];

assign p[4]= x[6] ^ y[4];
assign g[4]= x[6] & y[4];
assign c[5]= g[4] | (p[4] & g[3]) |(p[4] & p[3] & g[2]) |(p[4] & p[3] & p[2] & g[1]) | (p[4] & p[3] & p[2] & p[1] & g[0]) | (p[4] & p[3] & p[2] & p[1] & p[0] & cin);
assign sum[6]= p[4] ^ c[4];

assign p[5]= x[7] ^ y[5];
assign g[5]= x[7] & y[5];
assign c[6]= g[5] | (p[5] & g[4]) |(p[5] & p[4] & g[3]) |(p[5] & p[4] & p[3] & g[2]) | (p[5] & p[4] & p[3] & p[2] & g[1]) | (p[5] & p[4] & p[3] & p[2] & p[1] & g[0]) | (p[5] & p[4] & p[3] & p[2] & p[1] & p[0] & cin);
assign sum[7]= p[5] ^ c[5];

assign p[6]= x[8] ^ y[6];
assign g[6]= x[8] & y[6];
assign c[7]= g[6] | (p[6] & g[5]) | (p[6] & p[5] & g[4]) |(p[6] & p[5] & p[4] & g[3]) |(p[6] & p[5] & p[4] & p[3] & g[2]) | (p[6]  & p[5] & p[4] & p[3] & p[2] & g[1]) | (p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & g[0]) | (p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & p[0] & cin);
assign sum[8]= p[6] ^ c[6];

assign p[7]= x[9] ^ y[7];
assign g[7]= x[9] & y[7];
assign c[8]= g[7] | (p[7] & g[6]) | (p[7] & p[6] & g[5]) | (p[7] & p[6] & p[5] & g[4]) |(p[7] & p[6] & p[5] & p[4] & g[3]) |(p[7] & p[6] & p[5] & p[4] & p[3] & g[2]) | (p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & g[1]) | (p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & g[0]) | (p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & p[0] & cin);
assign sum[9]= p[7] ^ c[7];

assign p[8]= x[10] ^ y[8];
assign g[8]= x[10] & y[8];
assign c[9]= g[8] | (p[8] & g[7]) | (p[8] & p[7] & g[6]) | (p[8] & p[7] & p[6] & g[5]) | (p[8] & p[7] & p[6] & p[5] & g[4]) |(p[8] & p[7] & p[6] & p[5] & p[4] & g[3]) |(p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & g[2]) | (p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & g[1]) | (p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & g[0]) | (p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & p[0] & cin);
assign sum[10]= p[8] ^ c[8];

assign p[9]= x[11] ^ y[9];
assign g[9]= x[11] & y[9];
assign c[10]= g[9] | (p[9] & g[8]) | (p[9] & p[8] & g[7]) | (p[9] & p[8] & p[7] & g[6]) | (p[9] & p[8] & p[7] & p[6] & g[5]) | (p[9] & p[8] & p[7] & p[6] & p[5] & g[4]) |(p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & g[3]) |(p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & g[2]) | (p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & g[1]) | (p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & g[0]) | (p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & p[0] & cin);
assign sum[11]= p[9] ^ c[9];

assign p[10]= x[12] ^ y[10];
assign g[10]= x[12] & y[10];
assign c[11]= g[10] | (p[10] & g[9]) | (p[10] & p[9] & g[8]) | (p[10] & p[9] & p[8] & g[7]) | (p[10] & p[9] & p[8] & p[7] & g[6]) | (p[10] & p[9] & p[8] & p[7] & p[6] & g[5]) | (p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & g[4]) |(p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & g[3]) |(p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & g[2]) | (p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & g[1]) | (p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & g[0]) | (p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & p[0] & cin);
assign sum[12]= p[10] ^ c[10];

assign p[11]= x[13] ^ y[11];
assign g[11]= x[13] & y[11];
assign c[12]= g[11] | (p[11] & g[10]) | (p[11] & p[10] & g[9]) | (p[11] & p[10] & p[9] & g[8]) | (p[11] & p[10] & p[9] & p[8] & g[7]) | (p[11] & p[10] & p[9] & p[8] & p[7] & g[6]) | (p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & g[5]) | (p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & g[4]) |(p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & g[3]) |(p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & g[2]) | (p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & g[1]) | (p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & g[0]) | (p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & p[0] & cin);
assign sum[13]= p[11] ^ c[11];

assign p[12]= x[14] ^ y[12];
assign g[12]= x[14] & y[12];
assign c[13]= g[12] | (p[12] & g[11]) | (p[12] & p[11] & g[10]) | (p[12] & p[11] & p[10] & g[9]) | (p[12] & p[11] & p[10] & p[9] & g[8]) | (p[12] & p[11] & p[10] & p[9] & p[8] & g[7]) | (p[12] & p[11] & p[10] & p[9] & p[8] & p[7] & g[6]) | (p[12] & p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & g[5]) | (p[12] & p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & g[4]) |(p[12] & p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & g[3]) |(p[12] & p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & g[2]) | (p[12] & p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & g[1]) | (p[12] & p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & g[0]) | (p[12] & p[11] & p[10] & p[9] & p[8] & p[7] & p[6] & p[5] & p[4] & p[3] & p[2] & p[1] & p[0] & cin);
assign sum[14]= p[12] ^ c[12];

assign p[13]= x[15] ^ y[13];
assign g[13]= x[15] & y[13];
assign sum[15]= p[13] ^ c[13];

endmodule


Reference solution:
rtl/generated.v: 
assign uio_oe=0;




wire [7:0]mcand,mlier;
wire [15:0]ans;
latchh lat1(ui_in,clk,mcand);
latchh lat2(uio_in,clk,mlier);

wire [22:1]s;
wire [21:1]c;

wire [7:0]two_compl;
assign two_compl=~mcand+1;

reg [15:0]p0;
reg [13:0]p1;
reg [11:0]p2;
reg [9:0]p3;
 
wire [4:0]x1,x2,x3,x4;
//partial 1
 booth_encoding cod1(mlier[1],mlier[0],1'b0,x1[4],x1[3],x1[2],x1[1],x1[0]);
 always @( * )
        begin
          p0=16'd0;
          case (x1)
              5'b10000: p0=16'd0;
              5'b01000: p0={{8{mcand[7]}},mcand};
              5'b00100: p0={{8{two_compl[7]}},two_compl};
              5'b00010: p0={{7{mcand[7]}},mcand,1'b0};
              5'b00001: 
              begin
                p0={{7{two_compl[7]}},two_compl,1'b0};
               end
              default: p0=16'd0;
          endcase
      end   

//partial 2
    
    booth_encoding cod2(mlier[3],mlier[2],mlier[1],x2[4],x2[3],x2[2],x2[1],x2[0]);
 always @( * )
        begin
          p1=14'd0;
          case (x2)
              5'b10000: p1=14'd0;
              5'b01000: p1={{6{mcand[7]}},mcand};
              5'b00100: p1={{6{two_compl[7]}},two_compl};
              5'b00010: p1={{5{mcand[7]}},mcand,1'd0};
              5'b00001: 
              begin
                p1={{5{two_compl[7]}},two_compl,1'd0};
               end
              default: p1=14'd0;
          endcase
      end 
      
 //partial 3
 booth_encoding cod3(mlier[5],mlier[4],mlier[3],x3[4],x3[3],x3[2],x3[1],x3[0]);
 always @( * )
        begin
          p2=12'd0;
          case (x3)
              5'b10000: p2=12'd0;
              5'b01000: p2={{4{mcand[7]}},mcand};
              5'b00100: p2={{4{two_compl[7]}},two_compl};
              5'b00010: p2={{3{mcand[7]}},mcand,1'd0};
              5'b00001: 
              begin
                p2={{3{two_compl[7]}},two_compl,1'd0};
               end
              default: p2=12'd0;
          endcase
      end 
 
//partial 4
booth_encoding cod4(mlier[7],mlier[6],mlier[5],x4[4],x4[3],x4[2],x4[1],x4[0]);
always @( * )
        begin
          p3=10'd0;
          case (x4)
              5'b10000: p3=10'd0;
              5'b01000: p3={{2{mcand[7]}},mcand};
              5'b00100: p3={{2{two_compl[7]}},two_compl};
              5'b00010: p3={{1{mcand[7]}},mcand,1'd0};
              5'b00001: 
              begin
                p3={{1{two_compl[7]}},two_compl,1'd0};
               end
              default: p3=10'd0;
          endcase
      end 

//1st reduction 4 to 3
assign s[1]=p2[2] ^ p3[0];
assign c[1]=p2[2] & p3[0];

assign s[2]=(p1[5] ^p2[3]) ^ p3[1];
assign c[2]= (p1[5] & p2[3]) | (p2[3] & p3[1]) | (p3[1] & p1[5]);

assign s[3]=( p1[6] ^p2[4] ) ^ p3[2];
assign c[3]= (p1[6] & p2[4]) | (p2[4] & p3[2]) | (p3[2] & p1[6]);

assign s[4]=( p1[7] ^p2[5] ) ^ p3[3];
assign c[4]= (p1[7] & p2[5]) | (p2[5] & p3[3]) | (p3[3] & p1[7]);

assign s[5]=( p1[8] ^p2[6] ) ^ p3[4];
assign c[5]= (p1[8] & p2[6]) | (p2[6] & p3[4]) | (p3[4] & p1[8]);

assign s[6]=( p1[9] ^p2[7] ) ^ p3[5];
assign c[6]= (p1[9] & p2[7]) | (p2[7] & p3[5]) | (p3[5] & p1[9]);

assign s[7]=( p1[10] ^p2[8] ) ^ p3[6];
assign c[7]= (p1[10] & p2[8]) | (p2[8] & p3[6]) | (p3[6] & p1[10]);

assign s[8]=( p1[11] ^p2[9] ) ^ p3[7];
assign c[8]= (p1[11] & p2[9]) | (p2[9] & p3[7]) | (p3[7] & p1[11]);

assign s[9]=( p1[12] ^p2[10] ) ^ p3[8];
assign c[9]= (p1[12] & p2[10]) | (p2[10] & p3[8]) | (p3[8] & p1[12]);

assign s[10]=( p1[13] ^p2[11] ) ^ p3[9];
//assign c[10]= (p1[13] & p2[11]) | (p2[11] & p3[9]) | (p3[9] & p1[13]);

//2nd reduction 3 to 2
assign s[11]=p1[2] ^ p2[0];
assign c[11]=p1[2] & p2[0];

assign s[12]=( p0[5] ^p1[3] ) ^ p2[1];
assign c[12]= (p0[5] & p1[3]) | (p1[3] & p2[1]) | (p2[1] & p0[5]);

assign s[13]=( p0[6] ^p1[4] ) ^ s[1];
assign c[13]= (p0[6] & p1[4]) | (p1[4] & s[1]) | (s[1] & p0[6]);

assign s[14]=( p0[7] ^c[1] ) ^ s[2];
assign c[14]= (p0[7] & c[1]) | (c[1] & s[2]) | (s[2] & p0[7]);

assign s[15]=( p0[8] ^c[2] ) ^ s[3];
assign c[15]= (p0[8] & c[2]) | (c[2] & s[3]) | (s[3] & p0[8]);

assign s[16]=( p0[9] ^c[3] ) ^ s[4];
assign c[16]= (p0[9] & c[3]) | (c[3] & s[4]) | (s[4] & p0[9]);

assign s[17]=( p0[10] ^c[4] ) ^ s[5];
assign c[17]= (p0[10] & c[4]) | (c[4] & s[5]) | (s[5] & p0[10]);

assign s[18]=( p0[11] ^c[5] ) ^ s[6];
assign c[18]= (p0[11] & c[5]) | (c[5] & s[6]) | (s[6] & p0[11]);

assign s[19]=( p0[12] ^c[6] ) ^ s[7];
assign c[19]= (p0[12] & c[6]) | (c[6] & s[7]) | (s[7] & p0[12]);

assign s[20]=( p0[13] ^c[7] ) ^ s[8];
assign c[20]= (p0[13] & c[7]) | (c[7] & s[8]) | (s[8] & p0[13]);

assign s[21]=( p0[14] ^c[8] ) ^ s[9];
assign c[21]= (p0[14] & c[8]) | (c[8] & s[9]) | (s[9] & p0[14]);

assign s[22]=( p0[15] ^c[9] ) ^ s[10];


cla cla1(.x({c[21:11],p0[4:0]}),.y({s[22:11],p1[1],p1[0]}),.cin(1'b0),.sum({ans}));

latchh lat3(ans[7:0],clk,uo_out);
latchh lat4(ans[15:8],clk,uio_out);

  // List all unused inputs to prevent warnings
  wire _unused = &{ena, rst_n, 1'b0};