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]


// This file is part of OpenTSDB.
// Copyright (C) 2010-2012  The OpenTSDB Authors.
//
// This program is free software: you can redistribute it and/or modify it
// under the terms of the GNU Lesser General Public License as published by
// the Free Software Foundation, either version 2.1 of the License, or (at your
// option) any later version.  This program is distributed in the hope that it
// will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty
// of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser
// General Public License for more details.  You should have received a copy
// of the GNU Lesser General Public License along with this program.  If not,
// see <http://www.gnu.org/licenses/>.
package net.opentsdb.stats;

import java.util.Arrays;

/**
 * A histogram to keep track of the approximation of a distribution of values.
 * <p>
 * This is not a general purpose implementation of histogram.  It's
 * specifically designed for "small" values (close to 0) as the primary
 * use case is latency histograms.
 * <p>
 * All values must be positive ({@code >= 0}).
 * <p>
 * The histogram is linear (fixed size buckets) up to a given cutoff
 * point.  Beyond that point, the histogram becomes exponential (each
 * bucket is twice as large as the previous one).  This gives good
 * granularity for lower values while still allowing a rough
 * classification for the "long tail" of larger values.
 * <p>
 * Note that this implementation doesn't allow you to directly control
 * the number of buckets in the histogram.  The number will depend on
 * the arguments given to the constructor.
 * <p>
 * This class is not synchronized.
 */
public final class Histogram {

  /** Interval between each bucket for the linear part of the histogram. */
  private final short interval;

  /** Inclusive value beyond which we switch to exponential buckets. */
  private final int cutoff;

  /**
   * How many linear buckets we have.
   * Technically we don't need to store this value but we do in order to
   * avoid having to re-compute it in the fast path each time we add a
   * new value.
   */
  private final short num_linear_buckets;

  /**
   * The power of 2 used by the first exponential bucket.
   * Technically we don't need to store this value but we do in order to
   * avoid having to re-compute it in the fast path each time we add a
   * new value.
   */
  private final short exp_bucket_shift;

  /** Buckets where we actually store the values. */
  private final int[] buckets;

  /**
   * Constructor.
   * @param max The maximum value of the histogram.  Any value greater
   * than this will be considered to be "infinity".
   * @param interval The interval (size) of each linear bucket.
   * @param cutoff The value beyond which to switch to exponential
   * buckets.  The histogram may actually use this value or a value up
   * to {@code interval} greater.
   * @throws IllegalArgumentException if any of following conditions are
   * not met:
   * <pre>
   *   0 &lt; interval &lt;= max
   *   0 &lt;= cutoff &lt;= max
   * </pre>
   */
  public Histogram(final int max,
                   final short interval, final int cutoff) {
    if (interval > max) {
      throw new IllegalArgumentException("interval > max! interval="
                                         + interval + ", max=" + max);
    } else if (cutoff > max) {
      throw new IllegalArgumentException("cutoff > max! cutoff="
                                         + cutoff + ", max=" + max);
    } else if (interval < 1) {
      throw new IllegalArgumentException("interval < 1! interval=" + interval);
    } else if (cutoff < 0) {
      throw new IllegalArgumentException("cutoff < 0! interval=" + cutoff);
    }
    this.interval = interval;
    // One linear bucket every `interval' up to `cutoff'.
    num_linear_buckets = (short) (cutoff / interval);
    this.cutoff = num_linear_buckets * interval;
    this.exp_bucket_shift = (short) log2rounddown(interval);
    this.buckets = new int[num_linear_buckets
      // Find how many exponential buckets we need, starting from the
      // first power of 2 that's less than or equal to `interval'.
      + log2roundup((max - cutoff) >> exp_bucket_shift)
      // Add an extra overflow bucket at the end.
      + 1];
  }

  /**
   * Computes the logarithm base 2 (rounded up) of an integer.
   * <p>
   * This is essentially equivalent to
   *   {@code Math.ceil(Math.log(n) / Math.log(2))}
   * except it's 3 times faster.
   * @param n A strictly positive integer.
   * @return The logarithm base 2.  As a special case, if the integer
   * given in argument is 0, this function returns 0.  If the integer
   * given in argument is negative, the return value is undefined.
   * @see #log2rounddown
   */
  static final int log2roundup(final int n) {
    int log2 = 0;
    while (n > 1 << log2) {
      log2++;
    }
    return log2;
  }

  /**
   * Computes the logarithm base 2 (rounded down) of an integer.
   * <p>
   * This is essentially equivalent to
   *   {@code Math.floor(Math.log(n) / Math.log(2))}
   * except it's 4.5 times faster.  This function is also almost 70%
   * faster than {@link #log2roundup}.
   * @param n A strictly positive integer.
   * @return The logarithm base 2.  As a special case, if the integer
   * given in argument is 0, this function returns 0.  If the integer
   * given in argument is negative, the return value is undefined.
   * @see #log2roundup
   */
  static final int log2rounddown(int n) {
    int log2 = 0;
    while (n > 1) {
      n >>>= 1;
      log2++;
    }
    return log2;
  }

  /** Returns the number of buckets in this histogram. */
  public int buckets() {
    return buckets.length;
  }

  /**
   * Adds a value to the histogram.
   * <p>
   * This method works in {@code O(1)}.
   * @param value The value to add.
   * @throws IllegalArgumentException if the value given is negative.
   */
  public void add(final int value) {
    if (value < 0) {
      throw new IllegalArgumentException("negative value: " + value);
    }
    buckets[bucketIndexFor(value)]++;
  }

  /**
   * Returns the value of the <i>p</i>th  percentile in this histogram.
   * <p>
   * This method works in {@code O(N)} where {@code N} is the number of
   * {@link #buckets buckets}.
   * @param p A strictly positive integer in the range {@code [1; 100]}
   * @throws IllegalArgumentException if {@code p} is not valid.
   */
  public int percentile(int p) {
    if (p < 1 || p > 100) {
      throw new IllegalArgumentException("invalid percentile: " + p);
    }
    int count = 0;  // Count of values in the histogram.
    for (int i = 0; i < buckets.length; i++) {
      count += buckets[i];
    }
    if (count == 0) {  // Empty histogram.  Need to special-case it, otherwise
      return 0;        // the `if (count <= p)' below will be erroneously true.
    }
    // Find the number of elements at or below which the pth percentile is.
    p = count * p / 100;
    // Now walk the array backwards and decrement the count until it reaches p.
    for (int i = buckets.length - 1; i >= 0; i--) {
      count -= buckets[i];
      if (count <= p) {
        return bucketHighInterval(i);
      }
    }
    return 0;
  }

  /**
   * Prints this histogram in a human readable ASCII format.
   * <p>
   * This is equivalent to calling {@link #printAsciiBucket} on every
   * bucket.
   * @param out The buffer to which to write the output.
   */
  public void printAscii(final StringBuilder out) {
    for (int i = 0; i < buckets.length; i++) {
      printAsciiBucket(out, i);
    }
  }

  /**
   * Prints a bucket of this histogram in a human readable ASCII format.
   * @param out The buffer to which to write the output.
   * @see #printAscii
   */
  final void printAsciiBucket(final StringBuilder out, final int i) {
    out.append('[')
      .append(bucketLowInterval(i))
      .append('-')
      .append(i == buckets.length - 1 ? "Inf" : bucketHighInterval(i))
      .append("): ")
      .append(buckets[i])
      .append('\n');
  }

  /** Helper for unit tests that returns the value in the given bucket. */
  final int valueInBucket(final int index) {
    return buckets[index];
  }

  /** Finds the index of the bucket in which the given value should be. */
  private int bucketIndexFor(final int value) {
    if (value < cutoff) {
      return value / interval;
    }
    int bucket = num_linear_buckets  // Skip all linear buckets.
      // And find which bucket the rest (after `cutoff') should be in.
      // Reminder: the first exponential bucket ends at 2^exp_bucket_shift.
      + log2rounddown((value - cutoff) >> exp_bucket_shift);
    if (bucket >= buckets.length) {
      return buckets.length - 1;
    }
    return bucket;
  }

  /** Returns the low interval (inclusive) of the given bucket. */
  private int bucketLowInterval(final int index) {
    if (index <= num_linear_buckets) {
      return index * interval;
    } else {
      return cutoff + (1 << (index - num_linear_buckets + exp_bucket_shift));
    }
  }

  /** Returns the high interval (exclusive) of the given bucket. */
  private int bucketHighInterval(final int index) {
    if (index == buckets.length - 1) {
      return Integer.MAX_VALUE;
    } else {
      return bucketLowInterval(index + 1);
    }
  }

  public String toString() {
    return "Histogram(interval=" + interval + ", cutoff=" + cutoff
      + ", num_linear_buckets=" + num_linear_buckets
      + ", exp_bucket_shift=" + exp_bucket_shift
      + ", buckets=" + Arrays.toString(buckets) + ')';
  }

}