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 and FHT routines
**  Copyright 1988, 1993; Ron Mayer
**      Copyright (c) 1999-2000 Takehiro Tominaga
**
**  fht(fz,n);
**      Does a hartley transform of "n" points in the array "fz".
**
** NOTE: This routine uses at least 2 patented algorithms, and may be
**       under the restrictions of a bunch of different organizations.
**       Although I wrote it completely myself; it is kind of a derivative
**       of a routine I once authored and released under the GPL, so it
**       may fall under the free software foundation's restrictions;
**       it was worked on as a Stanford Univ project, so they claim
**       some rights to it; it was further optimized at work here, so
**       I think this company claims parts of it.  The patents are
**       held by R. Bracewell (the FHT algorithm) and O. Buneman (the
**       trig generator), both at Stanford Univ.
**       If it were up to me, I'd say go do whatever you want with it;
**       but it would be polite to give credit to the following people
**       if you use this anywhere:
**           Euler     - probable inventor of the fourier transform.
**           Gauss     - probable inventor of the FFT.
**           Hartley   - probable inventor of the hartley transform.
**           Buneman   - for a really cool trig generator
**           Mayer(me) - for authoring this particular version and
**                       including all the optimizations in one package.
**       Thanks,
**       <PRESIDIO_ANONYMIZED_PERSON>; <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>
** and added some optimization by
**           Mather    - idea of using lookup table
**           Takehiro  - some dirty hack for speed up
*/

/* $Id: fft.c,v 1.38 2009/04/20 21:48:00 robert Exp $ */

#ifdef HAVE_CONFIG_H
# include <config.h>
#endif

#include "lame.h"
#include "machine.h"
#include "encoder.h"
#include "util.h"
#include "fft.h"

#include "vector/lame_intrin.h"



#define TRI_SIZE (5-1)  /* 1024 =  4**5 */

/* fft.c    */
static FLOAT window[BLKSIZE], window_s[BLKSIZE_s / 2];

static const FLOAT costab[TRI_SIZE * 2] = {
    9.238795325112867e-01, 3.826834323650898e-01,
    9.951847266721969e-01, 9.801714032956060e-02,
    9.996988186962042e-01, 2.454122852291229e-02,
    9.999811752826011e-01, 6.135884649154475e-03
};

static void
fht(FLOAT * fz, int n)
{
    const FLOAT *tri = costab;
    int     k4;
    FLOAT  *fi, *gi;
    FLOAT const *fn;

    n <<= 1;            /* to get BLKSIZE, because of 3DNow! ASM routine */
    fn = fz + n;
    k4 = 4;
    do {
        FLOAT   s1, c1;
        int     i, k1, k2, k3, kx;
        kx = k4 >> 1;
        k1 = k4;
        k2 = k4 << 1;
        k3 = k2 + k1;
        k4 = k2 << 1;
        fi = fz;
        gi = fi + kx;
        do {
            FLOAT   f0, f1, f2, f3;
            f1 = fi[0] - fi[k1];
            f0 = fi[0] + fi[k1];
            f3 = fi[k2] - fi[k3];
            f2 = fi[k2] + fi[k3];
            fi[k2] = f0 - f2;
            fi[0] = f0 + f2;
            fi[k3] = f1 - f3;
            fi[k1] = f1 + f3;
            f1 = gi[0] - gi[k1];
            f0 = gi[0] + gi[k1];
            f3 = SQRT2 * gi[k3];
            f2 = SQRT2 * gi[k2];
            gi[k2] = f0 - f2;
            gi[0] = f0 + f2;
            gi[k3] = f1 - f3;
            gi[k1] = f1 + f3;
            gi += k4;
            fi += k4;
        } while (fi < fn);
        c1 = tri[0];
        s1 = tri[1];
        for (i = 1; i < kx; i++) {
            FLOAT   c2, s2;
            c2 = 1 - (2 * s1) * s1;
            s2 = (2 * s1) * c1;
            fi = fz + i;
            gi = fz + k1 - i;
            do {
                FLOAT   a, b, g0, f0, f1, g1, f2, g2, f3, g3;
                b = s2 * fi[k1] - c2 * gi[k1];
                a = c2 * fi[k1] + s2 * gi[k1];
                f1 = fi[0] - a;
                f0 = fi[0] + a;
                g1 = gi[0] - b;
                g0 = gi[0] + b;
                b = s2 * fi[k3] - c2 * gi[k3];
                a = c2 * fi[k3] + s2 * gi[k3];
                f3 = fi[k2] - a;
                f2 = fi[k2] + a;
                g3 = gi[k2] - b;
                g2 = gi[k2] + b;
                b = s1 * f2 - c1 * g3;
                a = c1 * f2 + s1 * g3;
                fi[k2] = f0 - a;
                fi[0] = f0 + a;
                gi[k3] = g1 - b;
                gi[k1] = g1 + b;
                b = c1 * g2 - s1 * f3;
                a = s1 * g2 + c1 * f3;
                gi[k2] = g0 - a;
                gi[0] = g0 + a;
                fi[k3] = f1 - b;
                fi[k1] = f1 + b;
                gi += k4;
                fi += k4;
            } while (fi < fn);
            c2 = c1;
            c1 = c2 * tri[0] - s1 * tri[1];
            s1 = c2 * tri[1] + s1 * tri[0];
        }
        tri += 2;
    } while (k4 < n);
}


static const unsigned char rv_tbl[] = {
    0x00, 0x80, 0x40, 0xc0, 0x20, 0xa0, 0x60, 0xe0,
    0x10, 0x90, 0x50, 0xd0, 0x30, 0xb0, 0x70, 0xf0,
    0x08, 0x88, 0x48, 0xc8, 0x28, 0xa8, 0x68, 0xe8,
    0x18, 0x98, 0x58, 0xd8, 0x38, 0xb8, 0x78, 0xf8,
    0x04, 0x84, 0x44, 0xc4, 0x24, 0xa4, 0x64, 0xe4,
    0x14, 0x94, 0x54, 0xd4, 0x34, 0xb4, 0x74, 0xf4,
    0x0c, 0x8c, 0x4c, 0xcc, 0x2c, 0xac, 0x6c, 0xec,
    0x1c, 0x9c, 0x5c, 0xdc, 0x3c, 0xbc, 0x7c, 0xfc,
    0x02, 0x82, 0x42, 0xc2, 0x22, 0xa2, 0x62, 0xe2,
    0x12, 0x92, 0x52, 0xd2, 0x32, 0xb2, 0x72, 0xf2,
    0x0a, 0x8a, 0x4a, 0xca, 0x2a, 0xaa, 0x6a, 0xea,
    0x1a, 0x9a, 0x5a, 0xda, 0x3a, 0xba, 0x7a, 0xfa,
    0x06, 0x86, 0x46, 0xc6, 0x26, 0xa6, 0x66, 0xe6,
    0x16, 0x96, 0x56, 0xd6, 0x36, 0xb6, 0x76, 0xf6,
    0x0e, 0x8e, 0x4e, 0xce, 0x2e, 0xae, 0x6e, 0xee,
    0x1e, 0x9e, 0x5e, 0xde, 0x3e, 0xbe, 0x7e, 0xfe
};

#define ch01(index)  (buffer[chn][index])

#define ml00(f) (window[i        ] * f(i))
#define ml10(f) (window[i + 0x200] * f(i + 0x200))
#define ml20(f) (window[i + 0x100] * f(i + 0x100))
#define ml30(f) (window[i + 0x300] * f(i + 0x300))

#define ml01(f) (window[i + 0x001] * f(i + 0x001))
#define ml11(f) (window[i + 0x201] * f(i + 0x201))
#define ml21(f) (window[i + 0x101] * f(i + 0x101))
#define ml31(f) (window[i + 0x301] * f(i + 0x301))

#define ms00(f) (window_s[i       ] * f(i + k))
#define ms10(f) (window_s[0x7f - i] * f(i + k + 0x80))
#define ms20(f) (window_s[i + 0x40] * f(i + k + 0x40))
#define ms30(f) (window_s[0x3f - i] * f(i + k + 0xc0))

#define ms01(f) (window_s[i + 0x01] * f(i + k + 0x01))
#define ms11(f) (window_s[0x7e - i] * f(i + k + 0x81))
#define ms21(f) (window_s[i + 0x41] * f(i + k + 0x41))
#define ms31(f) (window_s[0x3e - i] * f(i + k + 0xc1))


void
fft_short(lame_internal_flags const *const gfc,
          FLOAT x_real[3][BLKSIZE_s], int chn, const sample_t *const buffer[2])
{
    int     i;
    int     j;
    int     b;

    for (b = 0; b < 3; b++) {
        FLOAT  *x = &x_real[b][BLKSIZE_s / 2];
        short const k = (576 / 3) * (b + 1);
        j = BLKSIZE_s / 8 - 1;
        do {
            FLOAT   f0, f1, f2, f3, w;

            i = rv_tbl[j << 2];

            f0 = ms00(ch01);
            w = ms10(ch01);
            f1 = f0 - w;
            f0 = f0 + w;
            f2 = ms20(ch01);
            w = ms30(ch01);
            f3 = f2 - w;
            f2 = f2 + w;

            x -= 4;
            x[0] = f0 + f2;
            x[2] = f0 - f2;
            x[1] = f1 + f3;
            x[3] = f1 - f3;

            f0 = ms01(ch01);
            w = ms11(ch01);
            f1 = f0 - w;
            f0 = f0 + w;
            f2 = ms21(ch01);
            w = ms31(ch01);
            f3 = f2 - w;
            f2 = f2 + w;

            x[BLKSIZE_s / 2 + 0] = f0 + f2;
            x[BLKSIZE_s / 2 + 2] = f0 - f2;
            x[BLKSIZE_s / 2 + 1] = f1 + f3;
            x[BLKSIZE_s / 2 + 3] = f1 - f3;
        } while (--j >= 0);

        gfc->fft_fht(x, BLKSIZE_s / 2);
        /* BLKSIZE_s/2 because of 3DNow! ASM routine */
    }
}

void
fft_long(lame_internal_flags const *const gfc,
         FLOAT x[BLKSIZE], int chn, const sample_t *const buffer[2])
{
    int     i;
    int     jj = BLKSIZE / 8 - 1;
    x += BLKSIZE / 2;

    do {
        FLOAT   f0, f1, f2, f3, w;

        i = rv_tbl[jj];
        f0 = ml00(ch01);
        w = ml10(ch01);
        f1 = f0 - w;
        f0 = f0 + w;
        f2 = ml20(ch01);
        w = ml30(ch01);
        f3 = f2 - w;
        f2 = f2 + w;

        x -= 4;
        x[0] = f0 + f2;
        x[2] = f0 - f2;
        x[1] = f1 + f3;
        x[3] = f1 - f3;

        f0 = ml01(ch01);
        w = ml11(ch01);
        f1 = f0 - w;
        f0 = f0 + w;
        f2 = ml21(ch01);
        w = ml31(ch01);
        f3 = f2 - w;
        f2 = f2 + w;

        x[BLKSIZE / 2 + 0] = f0 + f2;
        x[BLKSIZE / 2 + 2] = f0 - f2;
        x[BLKSIZE / 2 + 1] = f1 + f3;
        x[BLKSIZE / 2 + 3] = f1 - f3;
    } while (--jj >= 0);

    gfc->fft_fht(x, BLKSIZE / 2);
    /* BLKSIZE/2 because of 3DNow! ASM routine */
}

#ifdef HAVE_NASM
extern void fht_3DN(FLOAT * fz, int n);
extern void fht_SSE(FLOAT * fz, int n);
#endif

void
init_fft(lame_internal_flags * const gfc)
{
    int     i;

    /* The type of window used here will make no real difference, but */
    /* in the interest of merging nspsytune stuff - switch to blackman window */
    for (i = 0; i < BLKSIZE; i++)
        /* blackman window */
        window[i] = 0.42 - 0.5 * cos(2 * PI * (i + .5) / BLKSIZE) +
            0.08 * cos(4 * PI * (i + .5) / BLKSIZE);

    for (i = 0; i < BLKSIZE_s / 2; i++)
        window_s[i] = 0.5 * (1.0 - cos(2.0 * PI * (i + 0.5) / BLKSIZE_s));

    gfc->fft_fht = fht;
#ifdef HAVE_NASM
    if (gfc->CPU_features.AMD_3DNow) {
        gfc->fft_fht = fht_3DN;
    }
    else if (gfc->CPU_features.SSE) {
        gfc->fft_fht = fht_SSE;
    }
    else {
        gfc->fft_fht = fht;
    }
#else
#ifdef HAVE_XMMINTRIN_H
#ifdef MIN_ARCH_SSE
    gfc->fft_fht = fht_SSE2;
#endif
#endif
#endif
}