You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


// ***********************************************************************
// fft_tester.c
// ***********************************************************************
//
// Read test vectors from fft_test_vectors.txt and send them to hardware
// Bitcoin wallet. The firmware should be compiled with the "TEST_FFT"
// preprocessor definition set.
//
// generate_test_vectors.m is a GNU Octave script which can
// be used to generate those test vectors. The test vectors compare the
// results of forward and inverse FFTs done by GNU Octave with FFTs done by
// the code in ../../fft.c.
//
// This also shows how much time (in clock cycles) each FFT required; this
// is useful for benchmarking.
//
// This file is licensed as described by the file LICENCE.

#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <string.h>

#include <unistd.h>
#include <fcntl.h>
#include <termios.h>

#include "../../fix16.h"
#include "../../fft.h" // include for FFT_SIZE

// The total relative error of the FFT result must be less than this in order
// for a test to pass. "Error" is the difference between the actual and
// expected output. Total relative error is the sum of errors for a FFT buffer
// divided by the sum of absolute values of the FFT result.
// This error metric is used because sometimes, if a FFT buffer contains
// a small value within lots of large values, that small value can have a
// large relative error ("transferred" to it from the large values during the
// FFT). This error metric takes into account the surrounding large values.
#define SUM_ERROR_THRESHOLD				0.001

// Every value in the FFT buffer must pass the absolute or relative error
// test (or both). If a single value fails both the absolute and relative
// error tests, then the entire FFT result is considered invalid.

// The absolute error test: the error must be lower than this.
// For Q16.16, this is 200 LSB.
// This is only a minimum; the actual allowable absolute error may be larger.
// Since the absolute error appears to scale as the standard deviation of
// the result array, the abolute error tolerance may be increased up to
// ERROR_FACTOR * standard deviation.
#define MINIMUM_ERROR_EPSILON			0.0030517578125
// The relative error test: the error divided by expected value must be
// lower than this. For values close to 0, the relative error can be huge;
// that's why there is also an absolute error test.
#define ERROR_FACTOR					0.0002


// The default number of bytes (transmitted or received) in between
// acknowledgments.
#define DEFAULT_ACKNOWLEDGE_INTERVAL	16
// The number of received bytes in between acknowledgments that this program
// will use (doesn't have to be the default).
#define RX_ACKNOWLEDGE_INTERVAL			32

// Remaining number of bytes that can be transmitted before listening for
// acknowledge.
static uint32_t tx_bytes_to_ack;
// Remaining number of bytes that can be received before other side expects an
// acknowledge.
static uint32_t rx_bytes_to_ack;

int fd_serial; // file descriptor for serial port

// Floating-point complex number.
typedef struct Complex_struct
{
	double real;
	double imag;
} Complex;

// Write the 32-bit unsigned integer specified by in into the byte array
// specified by out. This will write the bytes in a little-endian format.
static void writeU32LittleEndian(uint8_t *out, uint32_t in)
{
	out[0] = (uint8_t)in;
	out[1] = (uint8_t)(in >> 8);
	out[2] = (uint8_t)(in >> 16);
	out[3] = (uint8_t)(in >> 24);
}

// Read a 32-bit unsigned integer from the byte array specified by in.
// The bytes will be read in a little-endian format.
static uint32_t readU32LittleEndian(uint8_t *in)
{
	return ((uint32_t)in[0])
		| ((uint32_t)in[1] << 8)
		| ((uint32_t)in[2] << 16)
		| ((uint32_t)in[3] << 24);
}

// From fix16.h
static fix16_t fix16_from_dbl(double a)
{
	double temp = a * fix16_one;
#ifndef FIXMATH_NO_ROUNDING
	temp += (temp >= 0) ? 0.5f : -0.5f;
#endif
	return (fix16_t)temp;
}

// From fix16.h
static double fix16_to_dbl(fix16_t a)
{
	return (double)a / fix16_one;
}

// Get a byte from the serial link, sending an acknowledgement if required.
static uint8_t receiveByte(void)
{
	uint8_t ack_buffer[5];
	uint8_t buffer;

	read(fd_serial, &buffer, 1);
	rx_bytes_to_ack--;
	if (!rx_bytes_to_ack)
	{
		rx_bytes_to_ack = RX_ACKNOWLEDGE_INTERVAL;
		ack_buffer[0] = 0xff;
		writeU32LittleEndian(&(ack_buffer[1]), rx_bytes_to_ack);
		write(fd_serial, ack_buffer, 5);
	}
	return buffer;
}

// Send a byte to the serial link, waiting for acknowledgement if required.
static void sendByte(uint8_t data)
{
	uint8_t ack_buffer[5];
	uint8_t buffer;

	buffer = data;
	write(fd_serial, &buffer, 1);
	tx_bytes_to_ack--;
	if (!tx_bytes_to_ack)
	{
		read(fd_serial, ack_buffer, 5);
		if (ack_buffer[0] != 0xff)
		{
			printf("Unexpected acknowledgement format (%d)\n", (int)ack_buffer[0]);
			printf("Exiting, since the serial link is probably dodgy\n");
			exit(1);
		}
		tx_bytes_to_ack = readU32LittleEndian(&(ack_buffer[1]));
	}
}

// Read an array of double-precision real numbers from a file. Each number
// should be on a separate line.
static void readRealArray(FILE *f, double *array, uint32_t size)
{
	uint32_t i;

	for (i = 0; i < size; i++)
	{
		fscanf(f, "%lg\n", &(array[i]));
	}
}

// Read an array of double-precision complex numbers from a file. Each number
// should be on a separate line. The real components of every complex number
// in the array should be listed, followed by the imaginary components of
// every complex number.
// Why expect complex numbers in this format? Because it makes the GNU
// Octave script simpler.
static void readComplexArray(FILE *f, Complex *array, uint32_t size)
{
	uint32_t i;

	for (i = 0; i < size; i++)
	{
		fscanf(f, "%lg\n", &(array[i].real));
	}
	for (i = 0; i < size; i++)
	{
		fscanf(f, "%lg\n", &(array[i].imag));
	}
}

// Receive real number from serial link. The real number should be in Q16.16
// representation. This is so that the device under test doesn't have to
// do the conversion to floating-point.
static double receiveDouble(void)
{
	uint8_t buffer[4];
	int j;

	for (j = 0; j < 4; j++)
	{
		buffer[j] = receiveByte();
	}
	// The cast from uint32_t to fix16_t isn't platform-independent, because
	// the C99 specification doesn't make any guarantees about conversions
	// to signed integer types (...if the destination type cannot store the
	// source value, which will be the case if the fix16_t is negative).
	// But it should work on nearly every contemporary platform.
	return fix16_to_dbl((fix16_t)readU32LittleEndian(buffer));
}

// Send real number from serial link. The real number should be in Q16.16
// representation.
static void sendDouble(double value)
{
	uint8_t buffer[4];
	int j;

	if ((value > 32767.99998) || (value < -32767.99998))
	{
		printf("Tried to send \"%g\", which is outside the limits of fix16_t.\n", value);
		exit(1);
	}
	else
	{
		writeU32LittleEndian(buffer, (uint32_t)fix16_from_dbl(value));
		for (j = 0; j < 4; j++)
		{
			sendByte(buffer[j]);
		}
	}
}

// Receive complex number array from serial link. The numbers are expected to
// be interleaved: real[0], imaginary[0], real[1], imaginary[1] ...etc.
// This interleaving corresponds to how complex numbers are stored in memory.
static void receiveComplexArray(Complex *array, uint32_t size)
{
	uint32_t i;

	for (i = 0; i < size; i++)
	{
		array[i].real = receiveDouble();
		array[i].imag = receiveDouble();
	}
}

// Checks whether the given complex array represents a FFT error.
// The device under test signals a FFT error by sending an array consisting
// of all fix16_overflow.
static int isComplexArrayError(Complex *array, uint32_t size)
{
	uint32_t i;

	for (i = 0; i < size; i++)
	{
		if ((array[i].real != fix16_to_dbl(fix16_overflow))
			|| (array[i].imag != fix16_to_dbl(fix16_overflow)))
		{
			return 0;
		}
	}
	return 1;
}

// Send real array over serial link.
static void sendRealArray(double *array, uint32_t size)
{
	uint32_t i;

	for (i = 0; i < size; i++)
	{
		sendDouble(array[i]);
	}
}

// Send complex array over serial link, interleaving the real and imaginary
// components (as described above).
static void sendComplexArray(Complex *array, uint32_t size)
{
	uint32_t i;

	for (i = 0; i < size; i++)
	{
		sendDouble(array[i].real);
		sendDouble(array[i].imag);
	}
}

// Performs absolute and relative error tests.
// Returns 1 if at least one test passed, returns 0 if both tests failed.
// error_epsilon is the tolerance for the absolute error test.
// ERROR_FACTOR is the tolerance for the relative error test.
int equalWithinTolerance(double target, double value, double error_epsilon)
{
	double difference;

	difference = fabs(target - value);
	if (error_epsilon < MINIMUM_ERROR_EPSILON)
	{
		error_epsilon = MINIMUM_ERROR_EPSILON;
	}
	if (difference > error_epsilon)
	{
		if (target != 0.0)
		{
			if ((difference / fabs(target)) > ERROR_FACTOR)
			{
				return 0;
			}
		}
		else
		{
			return 0;
		}
	}
	return 1;
}

// Get standard deviation of the real or imaginary components of a complex
// array.
// is_imag = non-zero: get standard deviation of imaginary components.
// is_imag = zero: get standard deviation of real components.
static double getStandardDeviation(Complex *array, uint32_t size, int is_imag)
{
	double mean;
	double temp;
	double variance;
	uint32_t i;

	mean = 0.0;
	for (i = 0; i < size; i++)
	{
		if (is_imag)
		{
			mean += array[i].imag;
		}
		else
		{
			mean += array[i].real;
		}
	}
	mean /= (double)size;
	variance = 0.0;
	for (i = 0; i < size; i++)
	{
		if (is_imag)
		{
			temp = array[i].imag - mean;
		}
		else
		{
			temp = array[i].real - mean;
		}
		variance += temp * temp;
	}
	variance /= (double)size;
	return sqrt(variance);
}

// Check whether every value within a complex array matches every value within
// another complex array. This does an absolute and relative error test for
// each value, and also checks the total relative error.
// Returns 1 if all tests pass ("they are equal within error tolerance"), 0 if
// at least one test failed ("they are not equal within error tolerance").
int complexArraysEqualWithinTolerance(Complex *target, Complex *value, uint32_t size)
{
	uint32_t i;
	double error_sum;
	double target_size;
	double target_stdev_real;
	double target_stdev_imag;

	error_sum = 0;
	target_size = 0;
	target_stdev_real = getStandardDeviation(target, size, 0);
	target_stdev_imag = getStandardDeviation(target, size, 1);
	for (i = 0; i < size; i++)
	{
		if (!equalWithinTolerance(target[i].real, value[i].real, target_stdev_real * ERROR_FACTOR))
		{
			printf("%d.real mismatch: target = %g, value = %g ", i, target[i].real, value[i].real);
			return 0;
		}
		if (!equalWithinTolerance(target[i].imag, value[i].imag, target_stdev_imag * ERROR_FACTOR))
		{
			printf("%d.imag mismatch: target = %g, value = %g ", i, target[i].imag, value[i].imag);
			return 0;
		}
		error_sum += fabs(target[i].real - value[i].real);
		error_sum += fabs(target[i].imag - value[i].imag);
		target_size += fabs(target[i].real);
		target_size += fabs(target[i].imag);
	}
	if (target_size != 0)
	{
		error_sum /= target_size;
	}
	printf("err: %lg ", error_sum);
	if (error_sum > SUM_ERROR_THRESHOLD)
	{
		return 0;
	}
	else
	{
		return 1;
	}
}

int main(int argc, char **argv)
{
	int i;
	int j;
	int matches;
	int is_overflow_detection;
	int succeeded;
	int failed;
	char *newline_position;
	char buffer[512];
	uint8_t cycles_buffer[4];
	Complex input_normal[FFT_SIZE]; // input (normal-sized)
	Complex expected_normal[FFT_SIZE]; // expected output (normal-sized)
	Complex output_normal[FFT_SIZE]; // actual output (normal-sized)
	double input_double[FFT_SIZE * 2]; // input (double-sized)
	Complex expected_double[FFT_SIZE + 1]; // expected output (double-sized)
	Complex output_double[FFT_SIZE + 1]; // actual output (double-sized)
	FILE *f_vectors; // file containing test vectors
	struct termios options;
	struct termios old_options;

	if (argc != 2)
	{
		printf("Usage: %s <serial device>\n", argv[0]);
		printf("\n");
		printf("Example: %s /dev/ttyUSB0\n", argv[0]);
		exit(1);
	}

	// Attempt to open serial link.
	fd_serial = open(argv[1], O_RDWR | O_NOCTTY);
	if (fd_serial == -1)
	{
		printf("Could not open device \"%s\"\n", argv[1]);
		printf("Make sure you have permission to open it. In many systems, only\n");
		printf("root can access devices by default.\n");
		exit(1);
	}

	fcntl(fd_serial, F_SETFL, 0); // block on reads
	tcgetattr(fd_serial, &old_options); // save configuration
	memcpy(&options, &old_options, sizeof(options));
	cfsetispeed(&options, B57600); // baud rate 57600
	cfsetospeed(&options, B57600);
	options.c_cflag |= (CLOCAL | CREAD); // enable receiver and set local mode on
	options.c_cflag &= ~PARENB; // no parity
	options.c_cflag &= ~CSTOPB; // 1 stop bit
	options.c_cflag &= ~CSIZE; // character size mask
	options.c_cflag |= CS8; // 8 data bits
	options.c_lflag &= ~(ICANON | ECHO | ECHOE | ISIG); // raw input
	options.c_lflag &= ~(XCASE | ECHOK | ECHONL | ECHOCTL | ECHOPRT | ECHOKE); // disable more stuff
	options.c_iflag &= ~(IXON | IXOFF | IXANY); // no software flow control
	options.c_iflag &= ~(INPCK | INLCR | IGNCR | ICRNL | IUCLC); // disable more stuff
	options.c_oflag &= ~OPOST; // raw output
	tcsetattr(fd_serial, TCSANOW, &options);
	rx_bytes_to_ack = DEFAULT_ACKNOWLEDGE_INTERVAL;
	tx_bytes_to_ack = DEFAULT_ACKNOWLEDGE_INTERVAL;

	// Attempt to open file containing test vectors.
	f_vectors = fopen("fft_test_vectors.txt", "r");
	if (f_vectors == NULL)
	{
		printf("Could not open \"fft_test_vectors.txt\" for reading\n");
		exit(1);
	}

	succeeded = 0;
	failed = 0;
	while (!feof(f_vectors))
	{
		for (i = 0; i < 4; i++)
		{
			// Read name of test.
			fgets(buffer, sizeof(buffer), f_vectors);
			if (strstr(buffer, "overflow detection") != NULL)
			{
				is_overflow_detection = 1;
			}
			else
			{
				is_overflow_detection = 0;
			}
			// Remove newlines.
			newline_position = strrchr(buffer, '\n');
			if (newline_position != NULL)
			{
				*newline_position = '\0';
			}
			newline_position = strrchr(buffer, '\r');
			if (newline_position != NULL)
			{
				*newline_position = '\0';
			}
			printf("%s:\n    ", buffer);
			if (i < 2)
			{
				// Normal-sized tests.
				readComplexArray(f_vectors, input_normal, FFT_SIZE);
				readComplexArray(f_vectors, expected_normal, FFT_SIZE);
				sendComplexArray(input_normal, FFT_SIZE);
				receiveComplexArray(output_normal, FFT_SIZE);
				if (isComplexArrayError(output_normal, FFT_SIZE))
				{
					printf("FFT ERROR ");
					matches = is_overflow_detection;
				}
				else
				{
					matches = complexArraysEqualWithinTolerance(expected_normal, output_normal, FFT_SIZE);
				}
			}
			else
			{
				// Double-sized tests.
				readRealArray(f_vectors, input_double, FFT_SIZE * 2);
				readComplexArray(f_vectors, expected_double, FFT_SIZE + 1);
				sendRealArray(input_double, FFT_SIZE * 2);
				receiveComplexArray(output_double, FFT_SIZE + 1);
				if (isComplexArrayError(output_double, FFT_SIZE + 1))
				{
					printf("FFT ERROR ");
					matches = is_overflow_detection;
				}
				else
				{
					matches = complexArraysEqualWithinTolerance(expected_double, output_double, FFT_SIZE + 1);
				}
			}
			// Get number of cycles required to do FFT.
			for (j = 0; j < 4; j++)
			{
				cycles_buffer[j] = receiveByte();
			}
			printf("cycles = %u ", readU32LittleEndian(cycles_buffer));
			if (matches)
			{
				printf("[pass]\n");
				succeeded++;
			}
			else
			{
				printf("[fail]\n");
				// Make failure noticable.
				printf("************************\n");
				printf("FAIL FAIL FAIL FAIL FAIL\n");
				printf("************************\n");
				failed++;
			}
		}
	}

	printf("Tests which succeeded: %d\n", succeeded);
	printf("Tests which failed: %d\n", failed);

	fclose(f_vectors);
	tcsetattr(fd_serial, TCSANOW, &old_options); // restore configuration
	close(fd_serial);
	exit(0);
}