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 CL-123-abc
 * SPDX-License-Identifier: Apache-2.0
 */

/* Module name: tt_um_stochastic_addmultiply_CL123abc
 * Module description: 
 * Stochastic adder, multiplier and self-multiplier that takes in 9-bit (+ 1 bit buffer) inputs and gives 9-bit (+ 1 bit buffer) outputs
 * after 2^17+1 clock cycles. 
 * More details on the submodules are found after this module below.
 * INPUTS: 
 * ui_in[0] for serial input of 9bit (+1 bit buffer) probability.
 * ui_in[1] for serial input of 9bit (+1 bit buffer) probability.
 * The adder and multiplier take input from both inputs but the self-multiplier only takes input from ui_in[0].
 * OUTPUTS:
 * uo_out[0] for serial output of 9bit (+1 bit buffer) probability result of multiplier.
 * uo_out[1] for serial output of 9bit (+1 bit buffer) probability result of adder.
 * uo_out[2] for serial output of 9bit (+1 bit buffer) probability result of self-multiplier.
 * uo_out[3] signals the inner reset of the clk_counter of the module (not rst_n).
 */

`default_nettype none

module tt_um_stochastic_addmultiply_CL123abc(
    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 [8:0] input_1, input_2;
	wire [30:0] lfsr;
	wire SN_Bit_1, SN_Bit_1_Input2, SN_Bit_2, SN_Bit_sel;
	wire SN_Bit_mul_out, SN_Bit_add_out, SN_Bit_smul_out;
    wire [8:0] mul_avg, add_avg, smul_avg;
    wire mul_bit_out, add_bit_out, smul_bit_out;
    reg [17:0] clk_counter;
    
    serial_to_value_input global_input(.clk(clk), .clk_counter(clk_counter), .rst_n(rst_n), 
									   .input_bit_1(ui_in[0]), .output_bitseq_1(input_1), 
									   .input_bit_2(ui_in[1]), .output_bitseq_2(input_2));
	
	LFSR                 global_lsfr(.clk(clk), .rst_n(rst_n), .lfsr(lfsr));
	
	SN_Generators        global_SN_gen(.lfsr(lfsr), .Input_1(input_1), .Input_2(input_2), 
	                                   .SN_Bit_1(SN_Bit_1), .SN_Bit_1_Input2(SN_Bit_1_Input2), .SN_Bit_2(SN_Bit_2), 
	                                   .SN_Bit_sel(SN_Bit_sel));
	
	multiplier           mul(.SN_Bit_1(SN_Bit_1), .SN_Bit_2(SN_Bit_2), .SN_Bit_Out(SN_Bit_mul_out));
	adder                add(.SN_Bit_1(SN_Bit_1), .SN_Bit_2(SN_Bit_2), .SN_Bit_sel(SN_Bit_sel), .SN_Bit_Out(SN_Bit_add_out));
	self_multiplier      smul(.clk(clk), .SN_Bit_Input1(SN_Bit_1),.SN_Bit_Input2(SN_Bit_1_Input2), .SN_Bit_Out(SN_Bit_smul_out));
	
	up_counter           mul_up_counter(.clk(clk), .rst_n(rst_n), .SN_Bit_Out(SN_Bit_mul_out), 
	                                    .out_set(2'b00), .clk_counter(clk_counter), .average(mul_avg));
    up_counter           add_up_counter(.clk(clk), .rst_n(rst_n), .SN_Bit_Out(SN_Bit_add_out), 
                                        .out_set(2'b01), .clk_counter(clk_counter), .average(add_avg));
    up_counter           smul_up_counter(.clk(clk), .rst_n(rst_n), .SN_Bit_Out(SN_Bit_smul_out), 
                                         .out_set(2'b10), .clk_counter(clk_counter), .average(smul_avg));
                                         
    value_to_serial_output mul_output(.clk(clk), .rst_n(rst_n), .input_bits(mul_avg), .output_bit(mul_bit_out));
    value_to_serial_output add_output(.clk(clk), .rst_n(rst_n), .input_bits(add_avg), .output_bit(add_bit_out));
    value_to_serial_output smul_output(.clk(clk), .rst_n(rst_n), .input_bits(smul_avg), .output_bit(smul_bit_out));
  
	/* SEQUENTIAL LOGIC BLOCK:
	   The main code only controls the global clock counter and 
	   resets after 131072+1 or (2^17)+1 clk cycles counting 0th cycle.
	*/
	
    always @(posedge clk or posedge rst_n) begin
        	if (rst_n) begin
	    			clk_counter <= 0; 
        	end 
			else begin
        		if(clk_counter == 18'd131072)
        		    clk_counter <= 0;
        		else
        		    clk_counter <= clk_counter + 18'd1;
    		end 
	end  
  
  // PIN LAYOUT
  // All output pins must be assigned. If not used, assign to 0.
  /*
  assign uo_out[7:0] = mul_avg[8:1];
  assign uio_out[0] = mul_avg[0];
  assign uio_out[7:1] = 7'b0;
  assign uio_oe[7:0] = 8'b1;
  */
  
  assign uo_out[0] = mul_bit_out;
  assign uo_out[1] = add_bit_out; 
  assign uo_out[2] = smul_bit_out;
  assign uo_out[3] = clk_counter[17];
  assign uo_out[7:4] = 4'b0;
  assign uio_out[7:0] = 8'b0;    
  assign uio_oe[7:0]  = 8'b0;
  
  // List all unused inputs to prevent warnings
  wire _unused = &{ena, ui_in[7:2], uio_in, 1'b0}; 
endmodule

/* SUBMODULES:
     *
     * /////////////////////////////////////////////////////////////////////////////
	 * SUBMODULE NAME:
	   serial_to_value_input (.clk(), .clk_counter(), .rst_n(), 
	    					  .input_bit_1(), .output_bitseq_1(), 
	    					  .input_bit_2(), .output_bitseq_2());
     * SUBMODULE DESCRIPTION: 
     * Takes in serial input (10bit) carrying the 9bit probability in bipolar representation 
	 * and gives the 9bit value as output. The last bit in the 10 is the dummy bit.
     * INPUTS:
     * .clk() takes in the clk of the whole circuit.
	 * .clk_counter() takes in the global clock counter that is controlled by the main code.
     * .rst_n() takes in the reset of the whole circuit.
	 * .input_bit_1() takes in the serial bitstream of input 1, 1 bit per clk cycle. 
     * .input_bit_2() takes in the serial bitstream of input 2, 1 bit per clk cycle. 
     * OUTPUTS:
     * .output_bitseq_1() outputs the 9bit value of input 1.
	 * .output_bitseq_2() outputs the 9bit value of input 1.
     * \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
	 *
     * /////////////////////////////////////////////////////////////////////////////
	 * (UNUSED) SUBMODULE NAME:
	   input_checker (.input_precheck(), .output_postcheck());
     * SUBMODULE DESCRIPTION:
	 * Checks the input to see if it is within the desired limits. 
     * In this code it is only used for the self-multiplier, because the self-multiplier is scaled. 
	 * It cannot show result values if the input exceeds the limits. 
     * Currently, the limits are set to [9'b011110001 < input < 9'b100001111].
     * INPUTS:
     * .input_precheck() takes in a 9bit value to compare with the limit.
     * OUTPUTS:
	 * .output_postcheck() outputs a 9bit value that has already been cut to the limit.
     * \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
	 *
	 * /////////////////////////////////////////////////////////////////////////////
	 * SUBMODULE NAME:
       LFSR (.clk(), .rst_n(), .lfsr());
     * SUBMODULE DESCRIPTION:
	 * Generates and runs the LFSR, which is 31 bits long and has a seed of 31'd134995 in this code.
     * INPUTS:
     * .clk() takes in the clk of the whole circuit.
	 * .rst_n() takes in the reset of the whole circuit.
     * OUTPUTS:
	 * .lfsr() outputs the entire LFSR contents in 31-bit.
     * \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
	 *
	 * /////////////////////////////////////////////////////////////////////////////
	 * SUBMODULE NAME:
  	   SN_Generators (.lfsr(), .Input_1(), .Input_2(), 
	                  .SN_Bit_1(), .SN_Bit_1_Input2(), .SN_Bit_2(), 
	                  .SN_Bit_sel());
     * SUBMODULE DESCRIPTION:
	 * Generates stochastic bits by comparing the value from the input to the random number and generates a 1 if the input is larger.
     * The whole LFSR values is inputted so that different arrangements of its bits can result in more decorrelated random numbers.
     * INPUTS:
     * .lfsr() takes in the 31 bit LFSR values so that the random numbers can be extracted.
	 * .Input_1() takes in the 9bit bipolar probability value of input 1.
     * .Input_2() takes in the 9bit bipolar probability value of input 2.
     * OUTPUTS:
	 * .SN_Bit_1() outputs the generated SN bit from input 1, using its own lfsr value arrangement.
	 * .SN_Bit_1_Input2() outputs the generated SN bit from input 1, using the same lfsr value arrangement as SN_Bit_2.
     * .SN_Bit_2() outputs the generated SN bit from input 2, using the same lfsr value arrangement as SN_Bit_1_Input2.
	 * .SN_Bit_sel() outputs the generated SN bit from a unipolar probability value of 0.5, meant for use as the sel in the adder.
	 *               Uses its own lfsr value arrangement.
     * \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
	 *
     * /////////////////////////////////////////////////////////////////////////////
	 * SUBMODULE NAME:
  	   multiplier (.SN_Bit_1(), .SN_Bit_2(), .SN_Bit_Out());
     * SUBMODULE DESCRIPTION:
	 * Implements SN bipolar multiplication through a XNOR gate.
     * INPUTS:
     * .SN_Bit_1() takes in a SN bit with bipolar probability.
	 * .SN_Bit_2() takes in a SN bit with bipolar probability.
     * OUTPUTS:
	 * .SN_Bit_Out() outputs a SN bit with bipolar probability.
     * \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
	 *
     * /////////////////////////////////////////////////////////////////////////////
	 * SUBMODULE NAME:
  	   adder (.SN_Bit_1(), .SN_Bit_2(), .SN_Bit_sel(), .SN_Bit_Out());
     * SUBMODULE DESCRIPTION:
	 * Implements SN bipolar weighted averaging/addition through a MUX with SN_Bit_sel determining the weight of the inputs.
     * If SN_Bit_sel has a probability of 0.5, the result is averaging with equal weight between both inputs, 
	 * so it acts as an adder.
     * INPUTS:
     * .SN_Bit_1() takes in a SN bit with bipolar probability.
	 * .SN_Bit_2() takes in a SN bit with bipolar probability.
     * .SN_Bit_sel() takes in a SN bit with unipolar probability to determine the weight of the inputs.
     * OUTPUTS:
	 * .SN_Bit_Out() outputs a SN bit with bipolar probability.
     * \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
	 *
	 * /////////////////////////////////////////////////////////////////////////////
	 * (UNUSED) SUBMODULE NAME:
       absolute_value (.clk(), .SN_Bit_1(), .SN_Bit_Out());
     * SUBMODULE DESCRIPTION:
	 * This function was supposed to be the self-multiplier, but testing shows that it implements the 
	 * absolute function. 
	 * It takes in 1 input SN bitstream and uses a D-FF to generate what should be a 2nd decorrelated
	 * bitstream with the same probability, then passes them through XNOR gate for multiplication.
	 * Since it actually implements the absolute function, it seems that
	 * the decorrelation did not work here as intended. 
	 * Further testing should be done to figure out the correlation;
	 * this same function may also be able to be implemented without the D-FF.
     * INPUTS:
     * .clk() takes in the clk of the whole circuit.
	 * .SN_Bit_1() takes in a SN bit with bipolar probability.
     * OUTPUTS:
	 * .SN_Bit_Out() outputs a SN bit with bipolar probability.
     * \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
     * /////////////////////////////////////////////////////////////////////////////
	 * SUBMODULE NAME:
       self_multiplier (.clk(), .SN_Bit_Input1(),.SN_Bit_Input2(), .SN_Bit_Out());
     * SUBMODULE DESCRIPTION:
	 * This is the actual self_multiplier and it functions like the multiplier but the D-FF has been left in the bitstream.
	 * The 2 inputs are meant to take in the SN bits generated from the same input. SN_Bit_Input1 goes through the D-FF,
	 * but SN_Bit_Input2 does not. The 2 bits are then passed through an XNOR gate for multiplication.
	 * THe D-FF does not need to be included for multiplication, which can be seen in multiplier.
     * INPUTS:
     * .clk()takes in the clk of the whole circuit.
     * .SN_Bit_Input1()takes in a SN bit with bipolar probability, meant to be from the same input value.
     * .SN_Bit_Input2()takes in a SN bit with bipolar probability, meant to be from the same input value.
     * OUTPUTS:
	 * .SN_Bit_Out() outputs a SN bit with bipolar probability.
     * \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
	 *
     * /////////////////////////////////////////////////////////////////////////////
	 * SUBMODULE NAME:
       D_FF(.clk(), .D(), .Q());
     * SUBMODULE DESCRIPTION:
	 * D-flip flop. Takes D value and outputs it as Q every clk cycle.
     * INPUTS:
     * .clk() takes in the clk of the whole circuit.
	 * .D() takes in 1 bit.
     * OUTPUTS:
	 * .Q() takes in 1 bit.
     * \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
	 *
     * /////////////////////////////////////////////////////////////////////////////
	 * SUBMODULE NAME:
       up_counter (.clk(), .rst_n(), .SN_Bit_Out(), 
	               .out_set(), .clk_counter(), .average());
     * SUBMODULE DESCRIPTION:
	 * Counts the number of bits in a certain number of clk cycles and outputs the 9 MSB as the bipolar probability of 1s in the SN bitstream.
     * This code counts up to 2^17 + 1 clk cycles before outputting.
	 * out_set was meant to control scaling, but doesn't do anything currently.
     * INPUTS:
     * .clk() takes in the clk of the whole circuit.
	 * .rst_n() takes in the rst_n of the whole circuit.
     * .SN_Bit_Out() takes in the SN bitstream to be counted.
	 * .out_set() takes in a 2bit value to determine the case for the set of bits from the counter to be output.
     * 			  Currently, all options (2'b00, 2'b01, 2'b10) set the 9 MSB for output.
     * .clk_counter() takes in the global clock counter that is controlled by the main code.
     * OUTPUTS:
	 * .average() outputs a 9bit value, here that is the bipolar probability of 1s in the SN bitstream.
     * \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
	 *
     * /////////////////////////////////////////////////////////////////////////////
	 * SUBMODULE NAME:
       value_to_serial_output mul_output(.clk(), .rst_n(), .input_bits(), .output_bit());
     * SUBMODULE DESCRIPTION:
	 * Outputs each bit in the 9bit input as a serial bitstream with an added 10th bit as 0 as a dummy bit.
     * INPUTS:
     * .clk() takes in the clk of the whole circuit.
     * .rst_n() takes in the rst_n of the whole circuit.
	 * .input_bits() takes in a 9bit value, here that is the bipolar probability of 1s in the SN bitstream.
     * OUTPUTS:
	 * .output_bit() outputs each bit in the 9bit value and then outputs 0 for the 10th bit.
     * \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
	 *
 
*/

module serial_to_value_input(clk, clk_counter, rst_n, input_bit_1, output_bitseq_1, input_bit_2, output_bitseq_2);
input wire [17:0] clk_counter;
input wire clk, rst_n, input_bit_1, input_bit_2;
output reg [8:0] output_bitseq_1, output_bitseq_2; 
reg [8:0] output_bitcounter_1, output_bitcounter_2;
reg loop; 
reg [3:0] output_case;
reg [4:0] adjustment;

	always @(posedge clk or posedge rst_n) begin
    	if(rst_n) begin 
    		output_bitseq_1 <= 9'b0;
    		output_bitseq_2 <= 9'b0;
    		output_bitcounter_1 <= 9'b0;
    		output_bitcounter_2 <= 9'b0;
    		loop <= 1'b0;
			output_case <= 4'b0;
    		adjustment <= 5'd9;
    	end
    	else begin
			if (loop == 0) begin
				if (clk_counter == 0) begin
					case (output_case)
                    4'd0: adjustment <= 5'd9;
                    4'd1: adjustment <= 5'd16;
                    4'd2: adjustment <= 5'd13;
                    4'd3: adjustment <= 5'd10; 
                    4'd4: adjustment <= 5'd17; 
                    4'd5: adjustment <= 5'd14;
                    4'd6: adjustment <= 5'd11;   
                    4'd7: adjustment <= 5'd18;   
                    4'd8: adjustment <= 5'd17; 
                    4'd9: adjustment <= 5'd12;   
                    default:;
					endcase
				end
				output_bitcounter_1 <= (output_bitcounter_1 >> 1);
				output_bitcounter_1[8] <= input_bit_1;
				output_bitcounter_2 <= (output_bitcounter_2 >> 1);
				output_bitcounter_2[8] <= input_bit_2;
				if(clk_counter[4:0] == adjustment) begin
					output_bitseq_1 <= output_bitcounter_1;
					output_bitseq_2 <= output_bitcounter_2;
					loop <= 1;
				end
			end
			else if (loop == 1) begin
				if (clk_counter == 18'd131072) begin
					if(output_case == 4'd9)
						output_case <= 4'd0;
					else 
						output_case <= output_case + 4'd1;
				loop <= 0;
				end
			end
		end
	end
endmodule

/* UNUSED
module input_checker(input_precheck, output_postcheck); 
input wire [8:0] input_precheck;
wire over_limit, under_limit, limit;
output wire [8:0] output_postcheck;

assign over_limit = (input_precheck > 9'b100001111);
assign under_limit = (input_precheck < 9'b011110001);
assign limit = (over_limit||under_limit) == 1;
assign output_postcheck = (limit == 1)? ((over_limit == 1)? 9'b100001111 : 9'b011110001) : input_precheck;
endmodule
*/

module LFSR(clk, rst_n, lfsr);
// >>> Module Implementation Begin

// <<< Module Implementation End
endmodule



module SN_Generators(lfsr, Input_1, Input_2, SN_Bit_1, SN_Bit_1_Input2, SN_Bit_2, SN_Bit_sel);
input wire [30:0] lfsr;
input wire [8:0] Input_1, Input_2;
output wire SN_Bit_1, SN_Bit_1_Input2, SN_Bit_2, SN_Bit_sel;

assign SN_Bit_1 = (lfsr[8:0] < Input_1[8:0]) ;
assign SN_Bit_1_Input2 = (lfsr[20:12] < Input_1[8:0]);
assign SN_Bit_2 = (lfsr[20:12] < Input_2[8:0]) ;
assign SN_Bit_sel = ({lfsr[3:1], lfsr[30:26], lfsr[11]} < 9'b100000000);
wire unused = &{1'b0, lfsr[25:21], lfsr[10:9], 1'b0};
endmodule

module multiplier(SN_Bit_1, SN_Bit_2, SN_Bit_Out);
input wire SN_Bit_1, SN_Bit_2;
output wire SN_Bit_Out;
	
assign SN_Bit_Out = !(SN_Bit_1 ^ SN_Bit_2);
endmodule

module adder(SN_Bit_1, SN_Bit_2, SN_Bit_sel, SN_Bit_Out);
input wire SN_Bit_1, SN_Bit_2, SN_Bit_sel;
output wire SN_Bit_Out;
	
assign SN_Bit_Out = (SN_Bit_sel == 0) ? SN_Bit_1 : SN_Bit_2;
endmodule

/*UNUSED
module absolute_value(clk, SN_Bit_1, SN_Bit_Out);
input wire clk, SN_Bit_1;
output wire SN_Bit_Out; 
wire SN_Bit_Q;
D_FF delay_1_SN_Bit(clk, SN_Bit_1, SN_Bit_Q);

assign SN_Bit_Out = !(SN_Bit_1 ^ SN_Bit_Q);
endmodule
*/

module self_multiplier(clk, SN_Bit_Input1, SN_Bit_Input2, SN_Bit_Out);
input wire clk, SN_Bit_Input1, SN_Bit_Input2;
output wire SN_Bit_Out; 
wire SN_Bit_Q;
D_FF delay_1_SN_Bit(clk, SN_Bit_Input1, SN_Bit_Q);

assign SN_Bit_Out = !(SN_Bit_Q ^ SN_Bit_Input2);
endmodule

module D_FF(clk, D, Q);
input wire clk, D;
output reg Q;
	
	always@(posedge clk)begin
	       Q <= D;
	end
endmodule

module up_counter(clk, rst_n, SN_Bit_Out, out_set, clk_counter, average);
input wire clk, rst_n, SN_Bit_Out;
input wire [1:0] out_set;
input wire [17:0] clk_counter;
output reg [8:0] average;
reg [16:0] prob_counter;
//reg over_flag;

	always@(posedge clk or posedge rst_n) begin
		if (rst_n) begin
            average <= 0;
            prob_counter <= 0;
            //over_flag <= 0;
        end 
    	else begin
        	if (SN_Bit_Out == 1) begin
				if (prob_counter == 17'd131071) begin
		            //over_flag <= 1; 
		            prob_counter <= 17'b0;
	            end
	            else 
					prob_counter <= prob_counter + 17'b1;
	        end 
	        if (clk_counter == 18'd131072) begin 
				case(out_set)                           //For scaling, if needed.
			    2'b00: average <= {prob_counter[16:8]}; //Multiplier
			    2'b01: average <= {prob_counter[16:8]}; //Adder 
			    2'b10: average <= {prob_counter[16:8]}; //Self-Multiplier
			    default:;
	    		endcase
	    		//over_flag <= 0; 
	    		prob_counter <= 17'b0; 
	    	end
	    end
	end
endmodule

module value_to_serial_output(clk, rst_n, input_bits, output_bit);
input wire clk, rst_n;
input wire [8:0] input_bits;
reg [8:0] bitseq;
reg [3:0] counter;
output reg output_bit;

	always@(posedge clk or posedge rst_n) begin
		if(rst_n) begin
            bitseq <= 9'b0;
            counter <= 4'b0;
            output_bit <= 1'b0;
        end
        else begin
			if (counter == 0) begin
            	output_bit <= input_bits[0];
                bitseq <= input_bits >> 1;
                counter <= counter + 4'b1;
            end
			if (counter == 4'd9) begin
                output_bit <= 0;
                counter <= 4'b0;
            end
			else if (counter != 0 && counter != 4'd9) begin
                bitseq <= bitseq >> 1;
                output_bit <= bitseq[0];
                counter <= counter + 4'b1;
            end
         end
    end
endmodule


Reference solution:
rtl/generated.v: input wire clk, rst_n;
output reg [30:0] lfsr;
	always@(posedge clk or posedge rst_n) begin
        if (rst_n == 1)
            lfsr <= 31'd134995; // LFSR SEED AFFECTS ACCURACY A LOT
        else begin
            lfsr[0] <= lfsr[27] ^ lfsr[30] ;
            lfsr[30:1] <= lfsr[29:0] ;
		end
    end