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 Johannes Hoff
 * SPDX-License-Identifier: Apache-2.0
 */

`default_nettype none

module tt_um_johshoff_metaballs (
    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 rgb, display;
  wire[9:0] x;
  wire[9:0] y;

  wire h_sync;
  wire v_sync;
  assign uo_out = {h_sync, rgb, rgb, rgb, v_sync, rgb, rgb, rgb};
  assign uio_out = 0;
  assign uio_oe = 0;

  metaballs mb(
    rgb, v_sync,
    display,
    x,
    y
  );

  vga v(
    clk,
    ~rst_n,
    h_sync, v_sync,
    display,
    x,
    y
  );

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

endmodule


`default_nettype none

module vga(
	input wire clk_50mhz,
	input wire reset,
	output reg h_sync,
	output reg v_sync,

	output wire display,
	output wire[9:0] x,
	output reg[9:0] y
);
// >>> Module Implementation Begin

// <<< Module Implementation End
endmodule



module ball
#(
	parameter START_X = 30,
	parameter START_Y = 20,
	parameter BALL_SPEED = 5
)
(
	input wire[9:0] x,
	input wire[9:0] y,
	output wire[14:0] dist_sq,
	output wire overflow,

	input wire v_sync
);
	parameter SCREEN_WIDTH    = 800,
		      SCREEN_HEIGHT   = 600,
		      BALL_DIM        = 25;

	reg[9:0] ball_x = BALL_SPEED*START_X;
	reg[9:0] ball_y = BALL_SPEED*START_Y;
	reg ball_vx = 1;
	reg ball_vy = 1;

	wire[9:0] dx = x > ball_x ? x-ball_x : ball_x-x;
	wire[9:0] dy = y > ball_y ? y-ball_y : ball_y-y;

	wire[3:0] bit_count_x =
		  dx[9] == 1 ? 10
		: dx[8] == 1 ? 9
		: dx[7] == 1 ? 8
		: dx[6] == 1 ? 7
		: dx[5] == 1 ? 6
		: dx[4] == 1 ? 5
		: dx[3] == 1 ? 4
		: dx[2] == 1 ? 3
		: dx[1] == 1 ? 2
		: 0;

	wire[3:0] bit_count_y =
		  dy[9] == 1 ? 10
		: dy[8] == 1 ? 9
		: dy[7] == 1 ? 8
		: dy[6] == 1 ? 7
		: dy[5] == 1 ? 6
		: dy[4] == 1 ? 5
		: dy[3] == 1 ? 4
		: dy[2] == 1 ? 3
		: dy[1] == 1 ? 2
		: 0;

	wire overflow_x = bit_count_x + bit_count_x > 14;
	wire overflow_y = bit_count_y + bit_count_y > 14;

	// 800*800+600*600 = 1_000_000
	// log2 = 19.9
	// needs 20 bits
	//wire[14:0] dist_sq = dx*dx + dy*dy;
	assign dist_sq = dx*dx + dy*dy;
	assign overflow = overflow_x || overflow_y;

	always @(posedge v_sync) begin
		if (ball_vy) ball_y <= ball_y + BALL_SPEED;
		else         ball_y <= ball_y - BALL_SPEED;

		if (ball_y == BALL_SPEED)       ball_vy <= 1;
		if (ball_y == SCREEN_HEIGHT-BALL_DIM-BALL_SPEED) ball_vy <= 0;

		if (ball_vx) ball_x <= ball_x + BALL_SPEED;
		else         ball_x <= ball_x - BALL_SPEED;

		ball_vx <=
			ball_x == BALL_SPEED ? 1
			: ball_x == SCREEN_WIDTH-BALL_DIM-BALL_SPEED ? 0
			: ball_vx;
	end

endmodule

module metaballs(
	output wire rgb,
	input wire v_sync,

	input wire display,
	input wire[9:0] x,
	input wire[9:0] y
);
	wire[14:0] dist_sq_0;
	wire overflow_0;
	ball b_0(x, y, dist_sq_0, overflow_0, v_sync);
	// wire[14:0] dist_sq_1;
	// wire overflow_1;
	// ball #(.START_X(10), .START_Y(50)) b_1(x, y, dist_sq_1, overflow_1, v_sync);

	reg pix = 0;
	always @(posedge x[0]) begin
		pix <= (!overflow_0 && dist_sq_0 < 625); // || (!overflow_1 && dist_sq_1 < 625);
	end

	assign rgb = display && pix;

endmodule


Reference solution:
rtl/generated.v: 	// H_TIME_* is in hundreds of microseconds, writen nn_mm, where
	// nn is the microsecond part, _ is substituting for a decimal
	// separator, and mm is the fractional microsecond.

	// 800x600 @ 72Hz
	parameter H_TIME_VISIBLE_AREA  =  8_00,
		      H_TIME_FRONT_PORCH   =    56,
		      H_TIME_SYNC_PULSE    =  1_20,
		      H_TIME_WHOLE_LINE    = 10_40,

		      V_LINES_VISIBLE_AREA =   600,
		      V_LINES_FRONT_PORCH  =    37,
		      V_LINES_SYNC_PULSE   =     6,
		      V_LINES_WHOLE_FRAME  =   666;

	reg[10:0] h_counter;
	reg h_display;
	reg v_display;
	assign display = h_display & v_display;
	assign x = h_counter[9:0];

	always @(posedge clk_50mhz) begin
		// 0.00000001s = 0.00001ms = 0.01us
		// (31_77+1) * 0.01 = 31.78us
		//if (h_counter == 31_77) begin

		if (reset) begin
			h_counter <= 0;
			h_display <= 1;
			v_display <= 1;
			y <= 0;
			h_sync <= 1;
			v_sync <= 1;
		end else begin
			if (h_counter == H_TIME_WHOLE_LINE-1) begin
				h_counter <= 0;
				if (y == V_LINES_WHOLE_FRAME-1) begin
					v_display <= 1; // begining of vertical display
					y <= 0;
				end else begin
					y <= y + 1;
				end

				if (y == V_LINES_VISIBLE_AREA-1) v_display <= 0; // all way display
				if (y == V_LINES_VISIBLE_AREA+V_LINES_FRONT_PORCH-1) v_sync <= 0; // v_sync start
				if (y == V_LINES_VISIBLE_AREA+V_LINES_FRONT_PORCH+V_LINES_SYNC_PULSE-1) v_sync <= 1; // v_sync end
			end else begin
				h_counter <= h_counter + 1;
			end

			if (h_counter == H_TIME_WHOLE_LINE-1) h_display <= 1; // beginning of horizontal display
			if (h_counter == H_TIME_VISIBLE_AREA-1) h_display <= 0; // all the way display

			if (h_counter == H_TIME_VISIBLE_AREA+H_TIME_FRONT_PORCH-1) h_sync <= 0; // start h_sync
			if (h_counter == H_TIME_VISIBLE_AREA+H_TIME_FRONT_PORCH+H_TIME_SYNC_PULSE-1) h_sync <= 1; // end   h_sync
		end
	end