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 2024 Jaeden Amero
 * SPDX-License-Identifier: Apache-2.0
 */

`default_nettype none

module tt_um_patater_demokit (
    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
);

  // VGA signals
  wire hsync;
  wire vsync;
  // Should we reg for the colors? No need, palette has reg for it.
  wire [1:0] R;
  wire [1:0] G;
  wire [1:0] B;
  wire display_on;
  wire [9:0] hpos;
  wire [9:0] vpos;
  wire [5:0] rrggbb;
  wire [7:0] waggle_color;

  wire [7:0] color;

  // TinyVGA PMOD
  assign uo_out = {hsync, B[0], G[0], R[0], vsync, B[1], G[1], R[1]};

  // All output pins must be assigned. If not used, assign to 0.
  assign uio_out = 0;
  assign uio_oe  = 0;

  hvsync_generator hvsync_gen(
     .clk(clk),
     .reset(~rst_n),
     .hsync(hsync),
     .vsync(vsync),
     .display_on(display_on),
     .hpos(hpos),
     .vpos(vpos)
  );

  palette palette_inst (
    .color(color),
    .rrggbb(rrggbb)
  );

  waggle waggle_demo (
    .vsync(vsync),
    .rst_n(rst_n),
    .frame(frame[7:0]),
    .hpos(hpos),
    .vpos(vpos),
    .dual(ui_in[0]),
    .h(ui_in[1]),
    .p0_off(ui_in[4:2]),
    .p1_off(ui_in[7:5]),
    .color(waggle_color)
  );

  // List all unused inputs to prevent warnings
  wire _unused = &{ena, ui_in[7:3], uio_in, 1'b0};

  assign color = waggle_color;

  assign R = display_on ? rrggbb[5:4] : 2'b00;
  assign G = display_on ? rrggbb[3:2] : 2'b00;
  assign B = display_on ? rrggbb[1:0] : 2'b00;

  reg [9:0] frame;
  always @(posedge vsync or negedge rst_n) begin
    if (~rst_n) begin
      frame <= 0;
    end else begin
      frame <= frame + 1;
    end
  end

endmodule

module waggle (vsync, rst_n, frame, hpos, vpos, h, dual, p0_off, p1_off, color);
// >>> Module Implementation Begin

// <<< Module Implementation End
endmodule



module sine (x, y, xout, yout);
  input wire[9:0] x;
  input wire[9:0] y;
  output wire[9:0] xout;
  output wire[9:0] yout;

  reg [9:0] sine_lut[255:0];

  wire [1:0] xquadrant;
  wire [1:0] yquadrant;
  wire [7:0] xaddr;
  wire [7:0] yaddr;

  assign xquadrant = x[9:8];
  assign yquadrant = y[9:8];

  // Mirror quadrants 1 and 3
  assign xaddr = (xquadrant == 2'd1 || xquadrant == 2'd3) ? 8'd255 - x[7:0] :
                 x[7:0];
  assign yaddr = (yquadrant == 2'd1 || yquadrant == 2'd3) ? 8'd255 - y[7:0] :
                 y[7:0];

  // Negate quadrants 2 and 3
  assign xout = (xquadrant == 2'd2 || xquadrant == 2'd3) ? -sine_lut[xaddr] :
                sine_lut[xaddr];
  assign yout = (yquadrant == 2'd2 || yquadrant == 2'd3) ? -sine_lut[yaddr] :
                sine_lut[yaddr];

  initial begin
    sine_lut[0] = 10'd0;
    sine_lut[1] = 10'd1;
    sine_lut[2] = 10'd3;
    sine_lut[3] = 10'd5;
    sine_lut[4] = 10'd7;
    sine_lut[5] = 10'd9;
    sine_lut[6] = 10'd11;
    sine_lut[7] = 10'd13;
    sine_lut[8] = 10'd15;
    sine_lut[9] = 10'd17;
    sine_lut[10] = 10'd19;
    sine_lut[11] = 10'd21;
    sine_lut[12] = 10'd23;
    sine_lut[13] = 10'd25;
    sine_lut[14] = 10'd27;
    sine_lut[15] = 10'd29;
    sine_lut[16] = 10'd31;
    sine_lut[17] = 10'd33;
    sine_lut[18] = 10'd35;
    sine_lut[19] = 10'd37;
    sine_lut[20] = 10'd39;
    sine_lut[21] = 10'd41;
    sine_lut[22] = 10'd43;
    sine_lut[23] = 10'd45;
    sine_lut[24] = 10'd46;
    sine_lut[25] = 10'd48;
    sine_lut[26] = 10'd50;
    sine_lut[27] = 10'd52;
    sine_lut[28] = 10'd54;
    sine_lut[29] = 10'd56;
    sine_lut[30] = 10'd58;
    sine_lut[31] = 10'd60;
    sine_lut[32] = 10'd62;
    sine_lut[33] = 10'd64;
    sine_lut[34] = 10'd66;
    sine_lut[35] = 10'd68;
    sine_lut[36] = 10'd70;
    sine_lut[37] = 10'd72;
    sine_lut[38] = 10'd73;
    sine_lut[39] = 10'd75;
    sine_lut[40] = 10'd77;
    sine_lut[41] = 10'd79;
    sine_lut[42] = 10'd81;
    sine_lut[43] = 10'd83;
    sine_lut[44] = 10'd85;
    sine_lut[45] = 10'd87;
    sine_lut[46] = 10'd89;
    sine_lut[47] = 10'd91;
    sine_lut[48] = 10'd92;
    sine_lut[49] = 10'd94;
    sine_lut[50] = 10'd96;
    sine_lut[51] = 10'd98;
    sine_lut[52] = 10'd100;
    sine_lut[53] = 10'd102;
    sine_lut[54] = 10'd104;
    sine_lut[55] = 10'd105;
    sine_lut[56] = 10'd107;
    sine_lut[57] = 10'd109;
    sine_lut[58] = 10'd111;
    sine_lut[59] = 10'd113;
    sine_lut[60] = 10'd115;
    sine_lut[61] = 10'd116;
    sine_lut[62] = 10'd118;
    sine_lut[63] = 10'd120;
    sine_lut[64] = 10'd122;
    sine_lut[65] = 10'd124;
    sine_lut[66] = 10'd126;
    sine_lut[67] = 10'd127;
    sine_lut[68] = 10'd129;
    sine_lut[69] = 10'd131;
    sine_lut[70] = 10'd133;
    sine_lut[71] = 10'd135;
    sine_lut[72] = 10'd136;
    sine_lut[73] = 10'd138;
    sine_lut[74] = 10'd140;
    sine_lut[75] = 10'd142;
    sine_lut[76] = 10'd143;
    sine_lut[77] = 10'd145;
    sine_lut[78] = 10'd147;
    sine_lut[79] = 10'd149;
    sine_lut[80] = 10'd150;
    sine_lut[81] = 10'd152;
    sine_lut[82] = 10'd154;
    sine_lut[83] = 10'd156;
    sine_lut[84] = 10'd157;
    sine_lut[85] = 10'd159;
    sine_lut[86] = 10'd161;
    sine_lut[87] = 10'd162;
    sine_lut[88] = 10'd164;
    sine_lut[89] = 10'd166;
    sine_lut[90] = 10'd167;
    sine_lut[91] = 10'd169;
    sine_lut[92] = 10'd171;
    sine_lut[93] = 10'd172;
    sine_lut[94] = 10'd174;
    sine_lut[95] = 10'd176;
    sine_lut[96] = 10'd177;
    sine_lut[97] = 10'd179;
    sine_lut[98] = 10'd181;
    sine_lut[99] = 10'd182;
    sine_lut[100] = 10'd184;
    sine_lut[101] = 10'd185;
    sine_lut[102] = 10'd187;
    sine_lut[103] = 10'd189;
    sine_lut[104] = 10'd190;
    sine_lut[105] = 10'd192;
    sine_lut[106] = 10'd193;
    sine_lut[107] = 10'd195;
    sine_lut[108] = 10'd196;
    sine_lut[109] = 10'd198;
    sine_lut[110] = 10'd199;
    sine_lut[111] = 10'd201;
    sine_lut[112] = 10'd203;
    sine_lut[113] = 10'd204;
    sine_lut[114] = 10'd206;
    sine_lut[115] = 10'd207;
    sine_lut[116] = 10'd209;
    sine_lut[117] = 10'd210;
    sine_lut[118] = 10'd211;
    sine_lut[119] = 10'd213;
    sine_lut[120] = 10'd214;
    sine_lut[121] = 10'd216;
    sine_lut[122] = 10'd217;
    sine_lut[123] = 10'd219;
    sine_lut[124] = 10'd220;
    sine_lut[125] = 10'd222;
    sine_lut[126] = 10'd223;
    sine_lut[127] = 10'd224;
    sine_lut[128] = 10'd226;
    sine_lut[129] = 10'd227;
    sine_lut[130] = 10'd229;
    sine_lut[131] = 10'd230;
    sine_lut[132] = 10'd231;
    sine_lut[133] = 10'd233;
    sine_lut[134] = 10'd234;
    sine_lut[135] = 10'd235;
    sine_lut[136] = 10'd237;
    sine_lut[137] = 10'd238;
    sine_lut[138] = 10'd239;
    sine_lut[139] = 10'd241;
    sine_lut[140] = 10'd242;
    sine_lut[141] = 10'd243;
    sine_lut[142] = 10'd244;
    sine_lut[143] = 10'd246;
    sine_lut[144] = 10'd247;
    sine_lut[145] = 10'd248;
    sine_lut[146] = 10'd249;
    sine_lut[147] = 10'd251;
    sine_lut[148] = 10'd252;
    sine_lut[149] = 10'd253;
    sine_lut[150] = 10'd254;
    sine_lut[151] = 10'd255;
    sine_lut[152] = 10'd257;
    sine_lut[153] = 10'd258;
    sine_lut[154] = 10'd259;
    sine_lut[155] = 10'd260;
    sine_lut[156] = 10'd261;
    sine_lut[157] = 10'd262;
    sine_lut[158] = 10'd263;
    sine_lut[159] = 10'd264;
    sine_lut[160] = 10'd266;
    sine_lut[161] = 10'd267;
    sine_lut[162] = 10'd268;
    sine_lut[163] = 10'd269;
    sine_lut[164] = 10'd270;
    sine_lut[165] = 10'd271;
    sine_lut[166] = 10'd272;
    sine_lut[167] = 10'd273;
    sine_lut[168] = 10'd274;
    sine_lut[169] = 10'd275;
    sine_lut[170] = 10'd276;
    sine_lut[171] = 10'd277;
    sine_lut[172] = 10'd278;
    sine_lut[173] = 10'd279;
    sine_lut[174] = 10'd280;
    sine_lut[175] = 10'd281;
    sine_lut[176] = 10'd282;
    sine_lut[177] = 10'd283;
    sine_lut[178] = 10'd284;
    sine_lut[179] = 10'd284;
    sine_lut[180] = 10'd285;
    sine_lut[181] = 10'd286;
    sine_lut[182] = 10'd287;
    sine_lut[183] = 10'd288;
    sine_lut[184] = 10'd289;
    sine_lut[185] = 10'd290;
    sine_lut[186] = 10'd290;
    sine_lut[187] = 10'd291;
    sine_lut[188] = 10'd292;
    sine_lut[189] = 10'd293;
    sine_lut[190] = 10'd294;
    sine_lut[191] = 10'd294;
    sine_lut[192] = 10'd295;
    sine_lut[193] = 10'd296;
    sine_lut[194] = 10'd297;
    sine_lut[195] = 10'd297;
    sine_lut[196] = 10'd298;
    sine_lut[197] = 10'd299;
    sine_lut[198] = 10'd299;
    sine_lut[199] = 10'd300;
    sine_lut[200] = 10'd301;
    sine_lut[201] = 10'd301;
    sine_lut[202] = 10'd302;
    sine_lut[203] = 10'd303;
    sine_lut[204] = 10'd303;
    sine_lut[205] = 10'd304;
    sine_lut[206] = 10'd305;
    sine_lut[207] = 10'd305;
    sine_lut[208] = 10'd306;
    sine_lut[209] = 10'd306;
    sine_lut[210] = 10'd307;
    sine_lut[211] = 10'd307;
    sine_lut[212] = 10'd308;
    sine_lut[213] = 10'd308;
    sine_lut[214] = 10'd309;
    sine_lut[215] = 10'd309;
    sine_lut[216] = 10'd310;
    sine_lut[217] = 10'd310;
    sine_lut[218] = 10'd311;
    sine_lut[219] = 10'd311;
    sine_lut[220] = 10'd312;
    sine_lut[221] = 10'd312;
    sine_lut[222] = 10'd313;
    sine_lut[223] = 10'd313;
    sine_lut[224] = 10'd313;
    sine_lut[225] = 10'd314;
    sine_lut[226] = 10'd314;
    sine_lut[227] = 10'd314;
    sine_lut[228] = 10'd315;
    sine_lut[229] = 10'd315;
    sine_lut[230] = 10'd315;
    sine_lut[231] = 10'd316;
    sine_lut[232] = 10'd316;
    sine_lut[233] = 10'd316;
    sine_lut[234] = 10'd317;
    sine_lut[235] = 10'd317;
    sine_lut[236] = 10'd317;
    sine_lut[237] = 10'd317;
    sine_lut[238] = 10'd318;
    sine_lut[239] = 10'd318;
    sine_lut[240] = 10'd318;
    sine_lut[241] = 10'd318;
    sine_lut[242] = 10'd318;
    sine_lut[243] = 10'd318;
    sine_lut[244] = 10'd319;
    sine_lut[245] = 10'd319;
    sine_lut[246] = 10'd319;
    sine_lut[247] = 10'd319;
    sine_lut[248] = 10'd319;
    sine_lut[249] = 10'd319;
    sine_lut[250] = 10'd319;
    sine_lut[251] = 10'd319;
    sine_lut[252] = 10'd319;
    sine_lut[253] = 10'd319;
    sine_lut[254] = 10'd319;
    sine_lut[255] = 10'd319;
  end

endmodule

module palette (color, rrggbb);
  input wire [7:0] color;
  output wire [5:0] rrggbb;

  reg [5:0] palette;

  always @(color) begin
    case(1)
      color < 14: palette = 6'b110000;
      color >= 14 && color < 28: palette = 6'b110100;
      color >= 28 && color < 43: palette = 6'b111000;
      color >= 43 && color < 57: palette = 6'b111100;
      color >= 57 && color < 71: palette = 6'b101100;
      color >= 71 && color < 85: palette = 6'b011100;
      color >= 85 && color < 100: palette = 6'b001100;
      color >= 100 && color < 114: palette = 6'b001101;
      color >= 114 && color < 128: palette = 6'b001110;
      color >= 128 && color < 142: palette = 6'b001111;
      color >= 142 && color < 156: palette = 6'b001011;
      color >= 156 && color < 171: palette = 6'b000111;
      color >= 171 && color < 185: palette = 6'b000011;
      color >= 185 && color < 199: palette = 6'b010011;
      color >= 199 && color < 213: palette = 6'b100011;
      color >= 213 && color < 228: palette = 6'b110011;
      color >= 228 && color < 242: palette = 6'b110010;
      color >= 242: palette = 6'b110001;
    endcase
  end

  assign rrggbb = palette;
endmodule


`ifndef HVSYNC_GENERATOR_H
`define HVSYNC_GENERATOR_H

/*
Video sync generator, used to drive a VGA monitor.
Timing from: https://en.wikipedia.org/wiki/Video_Graphics_Array
To use:
- Wire the hsync and vsync signals to top level outputs
- Add a 3-bit (or more) "rgb" output to the top level
*/

module hvsync_generator(clk, reset, hsync, vsync, display_on, hpos, vpos);

  input clk;
  input reset;
  output reg hsync, vsync;
  output display_on;
  output reg [9:0] hpos;
  output reg [9:0] vpos;

  // declarations for TV-simulator sync parameters
  // horizontal constants
  parameter H_DISPLAY       = 640; // horizontal display width
  parameter H_BACK          =  48; // horizontal left border (back porch)
  parameter H_FRONT         =  16; // horizontal right border (front porch)
  parameter H_SYNC          =  96; // horizontal sync width
  // vertical constants
  parameter V_DISPLAY       = 480; // vertical display height
  parameter V_TOP           =  33; // vertical top border
  parameter V_BOTTOM        =  10; // vertical bottom border
  parameter V_SYNC          =   2; // vertical sync # lines
  // derived constants
  parameter H_SYNC_START    = H_DISPLAY + H_FRONT;
  parameter H_SYNC_END      = H_DISPLAY + H_FRONT + H_SYNC - 1;
  parameter H_MAX           = H_DISPLAY + H_BACK + H_FRONT + H_SYNC - 1;
  parameter V_SYNC_START    = V_DISPLAY + V_BOTTOM;
  parameter V_SYNC_END      = V_DISPLAY + V_BOTTOM + V_SYNC - 1;
  parameter V_MAX           = V_DISPLAY + V_TOP + V_BOTTOM + V_SYNC - 1;

  wire hmaxxed = (hpos == H_MAX) || reset;	// set when hpos is maximum
  wire vmaxxed = (vpos == V_MAX) || reset;	// set when vpos is maximum

  // horizontal position counter
  // XXX consider if we want hsync to be 0 or 1 on reset asserted
  always @(posedge clk)
  begin
    hsync <= (hpos>=H_SYNC_START && hpos<=H_SYNC_END);
    if(hmaxxed)
      hpos <= 0;
    else
      hpos <= hpos + 1;
  end

  // vertical position counter
  always @(posedge clk)
  begin
    vsync <= (vpos>=V_SYNC_START && vpos<=V_SYNC_END);
    if(hmaxxed)
      if (vmaxxed)
        vpos <= 0;
      else
        vpos <= vpos + 1;
  end

  // display_on is set when beam is in "safe" visible frame
  assign display_on = (hpos<H_DISPLAY) && (vpos<V_DISPLAY);

endmodule

`endif


Reference solution:
rtl/generated.v:   input wire vsync;
  input wire rst_n;
  input wire [7:0] frame;
  input wire [9:0] hpos;
  input wire [9:0] vpos;
  input wire h;
  input wire dual;
  input wire [2:0] p0_off;
  input wire [2:0] p1_off;
  output wire [7:0] color;

  wire signed [9:0] xsine_out;
  wire signed [9:0] ysine_out;

  wire [9:0] u;
  wire [9:0] v;
  // List all unused inputs to prevent warnings
  wire _unused = &{u[9:8], v[9:8]};

  reg [9:0] xpos;
  reg [9:0] ypos;
  reg [9:0] p0;
  reg [9:0] p1;

  always @(posedge vsync or negedge rst_n) begin
    if (~rst_n) begin
      xpos <= 0;
      ypos <= 0;
      p0 <= 0;
      p1 <= 0;
    end else begin
      p0 <= p0 + (10'd2 << p0_off);
      p1 <= p1 + (10'd3 << p1_off);
      xpos <= xpos + 10'd1;
      ypos <= ypos + 10'd2;
    end
  end

  sine xsine(
    .x(xsine_in),
    .y(ysine_in),
    .xout(xsine_out),
    .yout(ysine_out)
  );

  wire signed [9:0] xsine_in;
  wire signed [9:0] ysine_in;
  wire signed [9:0] xasine;
  wire signed [9:0] yasine;

  assign xsine_in = vpos + p0;
  assign xasine = xsine_out >>> 2;
  assign u = hpos + (dual ? (vpos[0] ? -xasine : xasine) : xasine) - xpos;

  assign ysine_in = (h ? hpos : vpos) + p1;
  assign yasine = ysine_out >>> 2;
  assign v = vpos + yasine - ypos;

  // Color with palette shift
  assign color = (u[7:0] ^ v[7:0]) + (frame << 1);