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]


1. Field of the Invention
The invention relates to computer systems in general and, more particularly, to computer methods and systems for solving sets of equations that can be represented in matrix form.
2. Description of the Related Art
Solving sets of equations is useful in, inter alia, mechanics, economics, fluid dynamics and electric circuit simulation. Circuit simulation, for example, typically involves assembling a set of equations that model the behavior of an electric circuit and then solving such set of equations to learn the value of unknown voltages and currents in the circuit. The equations that describe the circuit being simulated are often nonlinear and, for this reason, an iterative technique, such as the Newton Raphson method has typically been used to solve such set of equations. Various iterative techniques, including the Newton Raphson method, are well known to persons having ordinary skill in the art.
One part of the Newton Raphson method involves solving a set of linear equations. Empirically, the time needed to solve the set of equations dominates the time needed to simulate a large circuit. Using conventional circuit simulation techniques, even the most powerful supercomputers take many hours to complete a simulation and therefore, the simulation of large integrated circuits is still impractical.
Although a number of methods can be used to solve a set of linear equations, Lower Upper Decomposition ("LUD"), a method well known to persons having ordinary skill in the art, is generally preferred because of its accuracy and stability. LUD, however, cannot be parallelized easily.
A number of efforts have been made to parallelize LUD. J. Huang and O. Wing, "Optimal Parallel Triangulation of a Sparse Matrix," I.E.E.E. Trans. on Circuits and Systems, Vol. CAS-26, pp. 726-732 (Sept. 1979); N. Karmarkar, "A New Parallel Computer for Sparse Matrix Computation," Siam Conference on Discrete Mathematics, Atlanta (June 1990); and O. Wing and J. Huang, "A Computation Model of Parallel Solution of Linear Equations," I.E.E.E. Trans. on Computers, Vol. C-29, pp. 632-638 (July 1980). These methods concentrate on precompiling each step required in LUD and constructing a task graph which is then scheduled to run on a multiprocessor. These procedures yield a large number of tasks that need to be scheduled and require too much computer memory for realistic circuits.
An alternative approach, proposed by P. Sadayappan and V. Visvanathan, "Circuit Simulation on Shared Memory Multiprocessors," I.E.E.E. Transactions on Computers, Vol. C-37, pp. 1634-1642 (Dec. 1988), uses a partially compiled approach which has been shown to be very effective in extracting a high degree of parallelism from LUD of sparse matrices while minimizing the complexity of the task graph that has to be scheduled. This approach is a good compromise between the generic LUD approach and the compiled approach.
The Sadayappan-Visvanathan approach was originally developed for a shared memory multiprocessor and extended for use on a distributed memory multiprocessor in J. Trotter and P. Agrawal, "Circuit Simulation Methods on a Distributed Memory Multiprocessor System," I.E.E.E. Proceedings ICCAD, Santa Clara, pp. 438-441 (Nov. 1990). This reference is hereby incorporated by paper into this specification.
Another approach proposed by T. Nakata, N. Tanabe, H. Onozuka, T. Kurobe and N. Koike in "A Multiprocessor System for Modular Circuit Simulation," I.E.E.E. ICCAD, pp. 364-367 (1987), used LUD without compilation on a distributed memory multiprocessor. In the Nakata et. al. system each processor can access the memory of other processors using a shared bus.