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]


/******************************************************************
typeTruncatedLinear2D.h

Energy function with 2-dimensional truncated linear interactions:
   E(x)   =   \sum_i D_i(x_i)   +   \sum_ij V_ij(x_i,x_j)
   where x_i \in {0, 1, ..., KX-1} x {0, 1, ..., KY-1}
   (i.e. x_i = (x_i(1), x_i(2))).
   V_ij(ki, kj) = min { alpha1_ij*|ki(1)-kj(2)| + alpha2_ij*|ki(2)-kj(2)|, lambda_ij }.
   alpha1_ij, alpha2_ij and lambda_ij must be non-negative.

Example usage:

Minimize function E(x,y) = Dx(x) + Dy(y) + min { alpha1*|x(1) - y(1)| + alpha2*|x(2) - y(2)|, lambda } 
where 
  x,y \in {0,1} x {0,1,2}
  Dx(0,0) = 0, Dx(0,1) = 1, Dx(0,2) = 2,
  Dx(1,0) = 3, Dx(1,1) = 4, Dx(1,2) = 5,
  Dy(y) = 0 for all y,
  alpha1 = 6,
  alpha2 = 7,
  lambda = 8




#include <stdio.h>
#include "MRFEnergy.h"

void testTruncatedLinear2D()
{
	MRFEnergy<TypeTruncatedLinear2D>* mrf;
	MRFEnergy<TypeTruncatedLinear2D>::NodeId* nodes;
	MRFEnergy<TypeTruncatedLinear2D>::Options options;
	TypeTruncatedLinear2D::REAL energy, lowerBound;

	const int nodeNum = 2; // number of nodes
	const int KX = 2; //   label
	const int KY = 3; // dimensions
	const int K = KX*KY; // number of labels
	TypeTruncatedLinear2D::REAL D[K];
	TypeTruncatedLinear2D::Label x, y;

	mrf = new MRFEnergy<TypeTruncatedLinear2D>(TypeTruncatedLinear2D::GlobalSize(KX, KY));
	nodes = new MRFEnergy<TypeTruncatedLinear2D>::NodeId[nodeNum];

	// construct energy
	D[0] = 0; D[1] = 1; D[2] = 2;
	D[3] = 3; D[4] = 4; D[5] = 5;
	nodes[0] = mrf->AddNode(TypeTruncatedLinear2D::LocalSize(), TypeTruncatedLinear2D::NodeData(D));
	D[0] = 0; D[1] = 0; D[2] = 0;
	D[3] = 0; D[4] = 0; D[5] = 0;
	nodes[1] = mrf->AddNode(TypeTruncatedLinear2D::LocalSize(), TypeTruncatedLinear2D::NodeData(D));
	mrf->AddEdge(nodes[0], nodes[1], TypeTruncatedLinear2D::EdgeData(6, 7, 8));

	// Function below is optional - it may help if, for example, nodes are added in a random order
	// mrf->SetAutomaticOrdering();

	/////////////////////// TRW-S algorithm //////////////////////
	options.m_iterMax = 30; // maximum number of iterations
	mrf->Minimize_TRW_S(options, lowerBound, energy);

	// read solution
	x = mrf->GetSolution(nodes[0]);
	y = mrf->GetSolution(nodes[1]);

	printf("Solution: %d %d\n", x, y);

	//////////////////////// BP algorithm ////////////////////////
	mrf->ZeroMessages(); // in general not necessary - it may be faster to start 
	                     // with messages computed in previous iterations

	options.m_iterMax = 30; // maximum number of iterations
	mrf->Minimize_BP(options, energy);

	// read solution
	x = mrf->GetSolution(nodes[0]);
	y = mrf->GetSolution(nodes[1]);

	printf("Solution: (%d, %d) (%d, %d)\n", x.m_kx, x.m_ky, y.m_kx, y.m_ky);

	// done
	delete nodes;
	delete mrf;
}

*******************************************************************/


















#ifndef __TYPETRUNCATEDLINEAR2D_H__
#define __TYPETRUNCATEDLINEAR2D_H__

#include <string.h>
#include <assert.h>


template <class T> class MRFEnergy;


class TypeTruncatedLinear2D
{

public:
	// types declarations
	struct Edge; // stores edge information and either forward or backward message
	struct Vector; // node parameters and messages
	struct Label
	{
		int		m_kx, m_ky;
	};
	typedef double REAL;
	struct GlobalSize; // global information about number of labels
	struct LocalSize; // local information about number of labels (stored at each node)
	struct NodeData; // argument to MRFEnergy::AddNode()
	struct EdgeData; // argument to MRFEnergy::AddEdge()


	struct GlobalSize
	{
		GlobalSize(int KX, int KY);

	private:
	friend struct Vector;
	friend struct Edge;
		int		m_KX, m_KY; // label dimensions
		int		m_K; // number of labels
	};

	struct LocalSize // number of labels is stored at MRFEnergy::m_Kglobal
	{
	};

	struct NodeData
	{
		NodeData(REAL* data); // data = pointer to array of size MRFEnergy::m_Kglobal

	private:
	friend struct Vector;
	friend struct Edge;
		REAL*		m_data;
	};

	struct EdgeData
	{
		EdgeData(REAL alphaX, REAL alphaY, REAL lambda);

	private:
	friend struct Vector;
	friend struct Edge;
		REAL		m_alphaX, m_alphaY;
		REAL		m_lambda;
	};




	//////////////////////////////////////////////////////////////////////////////////
	////////////////////////// Visible only to MRFEnergy /////////////////////////////
	//////////////////////////////////////////////////////////////////////////////////


friend class MRFEnergy<TypeTruncatedLinear2D>;

	struct Vector
	{
		static int GetSizeInBytes(GlobalSize Kglobal, LocalSize K); // returns -1 if invalid K's
		void Initialize(GlobalSize Kglobal, LocalSize K, NodeData data);  // called once when user adds a node
		void Add(GlobalSize Kglobal, LocalSize K, NodeData data); // called once when user calls MRFEnergy::AddNodeData()

		void SetZero(GlobalSize Kglobal, LocalSize K);                            // set this[k] = 0
		void Copy(GlobalSize Kglobal, LocalSize K, Vector* V);                    // set this[k] = V[k]
		void Add(GlobalSize Kglobal, LocalSize K, Vector* V);                     // set this[k] = this[k] + V[k]
		REAL GetValue(GlobalSize Kglobal, LocalSize K, Label k);                  // return this[k]
		REAL ComputeMin(GlobalSize Kglobal, LocalSize K, Label& kMin);            // return vMin = min_k { this[k] }, set kMin
		REAL ComputeAndSubtractMin(GlobalSize Kglobal, LocalSize K);              // same as previous, but additionally set this[k] -= vMin (and kMin is not returned)

	private:
	friend struct Edge;
		REAL		m_data[1]; // actual size is MRFEnergy::m_Kglobal
	};

	struct Edge
	{
		static int GetSizeInBytes(GlobalSize Kglobal, LocalSize Ki, LocalSize Kj, EdgeData data); // returns -1 if invalid data
		static int GetBufSizeInBytes(int vectorMaxSizeInBytes); // returns size of buffer need for UpdateMessage()
		void Initialize(GlobalSize Kglobal, LocalSize Ki, LocalSize Kj, EdgeData data, Vector* Di, Vector* Dj); // called once when user adds an edge
		Vector* GetMessagePtr();
		void Swap(GlobalSize Kglobal, LocalSize Ki, LocalSize Kj); // if the client calls this function, then the meaning of 'dir'
								                                               // in distance transform functions is swapped

		// When UpdateMessage() is called, edge contains message from dest to source.
		// The function must replace it with the message from source to dest.
		// The update rule is given below assuming that source corresponds to tail (i) and dest corresponds
		// to head (j) (which is the case if dir==0).
		//
		// 1. Compute Di[ki] = gamma*source[ki] - message[ki].  (Note: message = message from j to i).
		// 2. Compute distance transform: set
		//       message[kj] = min_{ki} (Di[ki] + V(ki,kj)). (Note: message = message from i to j).
		// 3. Compute vMin = min_{kj} m_message[kj].
		// 4. Set m_message[kj] -= vMin.
		// 5. Return vMin.
		//
		// If dir==1 then source corresponds to j, sink corresponds to i. Then the update rule is
		//
		// 1. Compute Dj[kj] = gamma*source[kj] - message[kj].  (Note: message = message from i to j).
		// 2. Compute distance transform: set
		//       message[ki] = min_{kj} (Dj[kj] + V(ki,kj)). (Note: message = message from j to i).
		// 3. Compute vMin = min_{ki} m_message[ki].
		// 4. Set m_message[ki] -= vMin.
		// 5. Return vMin.
		//
		// If Edge::Swap has been called odd number of times, then the meaning of dir is swapped.
		//
		// Vector 'source' must not be modified. Function may use 'buf' as a temporary storage.
		REAL UpdateMessage(GlobalSize Kglobal, LocalSize Ksource, LocalSize Kdest, Vector* source, REAL gamma, int dir, void* buf);

		// If dir==0, then sets dest[kj] += V(ksource,kj).
		// If dir==1, then sets dest[ki] += V(ki,ksource).
		// If Swap() has been called odd number of times, then the meaning of dir is swapped.
		void AddColumn(GlobalSize Kglobal, LocalSize Ksource, LocalSize Kdest, Label ksource, Vector* dest, int dir);

	private:
		// edge information
		REAL		m_alphaX, m_alphaY;
		REAL		m_lambda;

		// message
		Vector		m_message;
	};
};




//////////////////////////////////////////////////////////////////////////////////
/////////////////////////////// Implementation ///////////////////////////////////
//////////////////////////////////////////////////////////////////////////////////


inline TypeTruncatedLinear2D::GlobalSize::GlobalSize(int KX, int KY)
{
	m_KX = KX;
	m_KY = KY;
	m_K = KX*KY;
}

///////////////////// NodeData and EdgeData ///////////////////////

inline TypeTruncatedLinear2D::NodeData::NodeData(REAL* data)
{
	m_data = data;
}

inline TypeTruncatedLinear2D::EdgeData::EdgeData(REAL alphaX, REAL alphaY, REAL lambda)
{
	m_alphaX = alphaX;
	m_alphaY = alphaY;
	m_lambda = lambda;
}

///////////////////// Vector ///////////////////////

inline int TypeTruncatedLinear2D::Vector::GetSizeInBytes(GlobalSize Kglobal, LocalSize K)
{
	if (Kglobal.m_KX < 1 || Kglobal.m_KY < 2)
	{
		return -1;
	}
	return Kglobal.m_K*sizeof(REAL);
}
inline void TypeTruncatedLinear2D::Vector::Initialize(GlobalSize Kglobal, LocalSize K, NodeData data)
{
	memcpy(m_data, data.m_data, Kglobal.m_K*sizeof(REAL));
}

inline void TypeTruncatedLinear2D::Vector::Add(GlobalSize Kglobal, LocalSize K, NodeData data)
{
	for (int k=0; k<Kglobal.m_K; k++)
	{
		m_data[k] += data.m_data[k];
	}
}

inline void TypeTruncatedLinear2D::Vector::SetZero(GlobalSize Kglobal, LocalSize K)
{
	memset(m_data, 0, Kglobal.m_K*sizeof(REAL));
}

inline void TypeTruncatedLinear2D::Vector::Copy(GlobalSize Kglobal, LocalSize K, Vector* V)
{
	memcpy(m_data, V->m_data, Kglobal.m_K*sizeof(REAL));
}

inline void TypeTruncatedLinear2D::Vector::Add(GlobalSize Kglobal, LocalSize K, Vector* V)
{
	for (int k=0; k<Kglobal.m_K; k++)
	{
		m_data[k] += V->m_data[k];
	}
}

inline TypeTruncatedLinear2D::REAL TypeTruncatedLinear2D::Vector::GetValue(GlobalSize Kglobal, LocalSize K, Label k)
{
	assert(k.m_kx>=0 && k.m_kx<Kglobal.m_KX && k.m_ky>=0 && k.m_ky<Kglobal.m_KY);
	return m_data[k.m_kx + k.m_ky*Kglobal.m_KX];
}

inline TypeTruncatedLinear2D::REAL TypeTruncatedLinear2D::Vector::ComputeMin(GlobalSize Kglobal, LocalSize K, Label& _kMin)
{
	REAL vMin = m_data[0];
	int kMin = 0;
	for (int k=1; k<Kglobal.m_K; k++)
	{
		if (vMin > m_data[k])
		{
			vMin = m_data[k];
			kMin = k;
		}
	}

	_kMin.m_ky = kMin / Kglobal.m_KX;
	_kMin.m_kx = kMin - _kMin.m_ky * Kglobal.m_KX;

	return vMin;
}

inline TypeTruncatedLinear2D::REAL TypeTruncatedLinear2D::Vector::ComputeAndSubtractMin(GlobalSize Kglobal, LocalSize K)
{
	REAL vMin = m_data[0];
	for (int k=1; k<Kglobal.m_K; k++)
	{
		if (vMin > m_data[k])
		{
			vMin = m_data[k];
		}
	}
	for (int k=0; k<Kglobal.m_K; k++)
	{
		m_data[k] -= vMin;
	}

	return vMin;
}

///////////////////// EdgeDataAndMessage implementation /////////////////////////

inline int TypeTruncatedLinear2D::Edge::GetSizeInBytes(GlobalSize Kglobal, LocalSize Ki, LocalSize Kj, EdgeData data)
{
	if (data.m_alphaX < 0 || data.m_alphaY < 0 || data.m_lambda < 0)
	{
		return -1;
	}
	return sizeof(Edge) - sizeof(Vector) + Kglobal.m_K*sizeof(REAL);
}

inline int TypeTruncatedLinear2D::Edge::GetBufSizeInBytes(int vectorMaxSizeInBytes)
{
	return 0;
}

inline void TypeTruncatedLinear2D::Edge::Initialize(GlobalSize Kglobal, LocalSize Ki, LocalSize Kj, EdgeData data, Vector* Di, Vector* Dj)
{
	m_alphaX = data.m_alphaX;
	m_alphaY = data.m_alphaY;
	m_lambda = data.m_lambda;
}

inline TypeTruncatedLinear2D::Vector* TypeTruncatedLinear2D::Edge::GetMessagePtr()
{
	return &m_message;
}

inline void TypeTruncatedLinear2D::Edge::Swap(GlobalSize Kglobal, LocalSize Ki, LocalSize Kj)
{
}

inline TypeTruncatedLinear2D::REAL TypeTruncatedLinear2D::Edge::UpdateMessage(GlobalSize Kglobal, LocalSize Ksource, LocalSize Kdest, Vector* source, REAL gamma, int dir, void* buf)
{
	Label k;
	REAL vMin;
	REAL* sourcePtr;
	REAL* destPtr;

	m_message.m_data[0] = gamma*source->m_data[0] - m_message.m_data[0];
	vMin = m_message.m_data[0];

	sourcePtr = source->m_data;
	destPtr = m_message.m_data;
	for (k.m_ky=0; k.m_ky<Kglobal.m_KY; k.m_ky++)
	for (k.m_kx=0; k.m_kx<Kglobal.m_KX; k.m_kx++, sourcePtr++, destPtr++)
	{
		destPtr[0] = gamma*sourcePtr[0] - destPtr[0];
		if (k.m_kx > 0 && destPtr[0] > destPtr[-1] + m_alphaX)
		{
			destPtr[0] = destPtr[-1] + m_alphaX;
		}
		if (k.m_ky > 0 && destPtr[0] > destPtr[-Kglobal.m_KX] + m_alphaY)
		{
			destPtr[0] = destPtr[-Kglobal.m_KX] + m_alphaY;
		}
		else if (vMin > destPtr[0])
		{
			vMin = destPtr[0];
		}
	}

	destPtr--;
	for (k.m_ky=Kglobal.m_KY-1; k.m_ky>=0; k.m_ky--)
	for (k.m_kx=Kglobal.m_KX-1; k.m_kx>=0; k.m_kx--, destPtr--)
	{
		destPtr[0] -= vMin;
		if (k.m_kx < Kglobal.m_KX-1 && destPtr[0] > destPtr[1] + m_alphaX)
		{
			destPtr[0] = destPtr[1] + m_alphaX;
		}
		if (k.m_ky < Kglobal.m_KY-1 && destPtr[0] > destPtr[Kglobal.m_KX] + m_alphaY)
		{
			destPtr[0] = destPtr[Kglobal.m_KX] + m_alphaY;
		}
		if (destPtr[0] > m_lambda)
		{
			destPtr[0] = m_lambda;
		}
	}

	return vMin;
}

inline void TypeTruncatedLinear2D::Edge::AddColumn(GlobalSize Kglobal, LocalSize Ksource, LocalSize Kdest, Label ksource, Vector* dest, int dir)
{
	assert(ksource.m_kx>=0 && ksource.m_kx<Kglobal.m_KX && ksource.m_ky>=0 && ksource.m_ky<Kglobal.m_KY);

	REAL* destPtr;
	Label k;

	destPtr = dest->m_data;
	for (k.m_ky=0; k.m_ky<Kglobal.m_KY; k.m_ky++)
	for (k.m_kx=0; k.m_kx<Kglobal.m_KX; k.m_kx++, destPtr++)
	{
		REAL cost = m_alphaX*((k.m_kx > ksource.m_kx) ? k.m_kx - ksource.m_kx : ksource.m_kx - k.m_kx)
		          + m_alphaY*((k.m_ky > ksource.m_ky) ? k.m_ky - ksource.m_ky : ksource.m_ky - k.m_ky);
		destPtr[0] += (cost < m_lambda) ? cost : m_lambda;
	}
}

//////////////////////////////////////////////////////////////////////////////////

#endif