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]


module MerkleTree.Low.VectorExtras

module B = LowStar.Buffer
module S = FStar.Seq
module HS = FStar.HyperStack
module HST = FStar.HyperStack.ST
module U32 = FStar.UInt32

open LowStar.BufferOps
open Hacl.Hash.Lemmas
open FStar.Integers
open LowStar.Modifies
open LowStar.Regional

open LowStar.Vector
open LowStar.RVector
module V = LowStar.Vector
module RV = LowStar.RVector


(** Some extra functions on top of LowStar.Vector... used for Merkle Tree. *)

#set-options "--max_fuel 0 --max_ifuel 0 --z3rlimit 50"

inline_for_extraction
let move_left #a (b: B.buffer a) (dst src: U32.t) (l: U32.t): HST.Stack unit
  (requires fun h0 ->
    B.live h0 b /\
    U32.v src + U32.v l <= B.length b /\
    U32.v dst <= U32.v src)
  (ensures fun h0 _ h1 ->
    B.(modifies (loc_buffer b) h0 h1) /\ (

    let b0 = B.as_seq h0 b in
    let b1 = B.as_seq h1 b in
    let src = U32.v src in
    let dst = U32.v dst in
    let l = U32.v l in
    S.slice b1 dst (dst + l) `S.equal` S.slice b0 src (src + l)))
=
  let h0 = HST.get () in
  [@inline_let]
  let inv (h: HS.mem) (i: nat) =
    let b0 = B.as_seq h0 b in
    let b1 = B.as_seq h b in
    let src = U32.v src in
    let dst = U32.v dst in
    let l = U32.v l in
    i <= l /\
    B.(modifies (loc_buffer b) h0 h) /\
    S.slice b1 dst (dst + i) `S.equal` S.slice b0 src (src + i) /\
    S.slice b1 (src + i) (src + l) `S.equal` S.slice b0 (src + i) (src + l)
  in
  let f (i: U32.t { U32.(0 <= v i /\ v i < v l) }): HST.Stack unit
    (requires fun h0 -> inv h0 (U32.v i))
    (ensures fun h0 _ h1 -> U32.(inv h0 (v i) /\ inv h1 (v i + 1)))
  =
    let h00 = HST.get () in
    calc (==) {
      S.index (B.as_seq h0 b) U32.(v src + v i);
    (==) {}
      S.index (S.slice (B.as_seq h0 b) U32.(v src + v i) U32.(v src + v l)) 0;
    (==) {}
      S.index (S.slice (B.as_seq h00 b) U32.(v src + v i) U32.(v src + v l)) 0;
    (==) {}
      S.index (B.as_seq h00 b) U32.(v src + v i);
    };
    b.(dst `U32.add` i) <- b.(src `U32.add` i);
    let h = HST.get () in
    let b0 = B.as_seq h0 b in
    let b1 = B.as_seq h b in
    let src = U32.v src in
    let dst = U32.v dst in
    let l = U32.v l in
    let i = U32.v i in
    calc (S.equal) {
      S.slice b1 dst (dst + (i + 1));
    (S.equal) { lemma_slice_ijk b1 dst (dst + i) (dst + i + 1) }
      S.slice b1 dst (dst + i) `S.append` S.slice b1 (dst + i) (dst + i + 1);
    (S.equal) { }
      S.slice b0 src (src + i) `S.append` S.slice b1 (dst + i) (dst + i + 1);
    (S.equal) { }
      S.slice b0 src (src + i) `S.append` S.cons (S.index b1 (dst + i)) S.empty;
    (S.equal) { }
      S.slice b0 src (src + i) `S.append` S.cons (S.index b0 (src + i)) S.empty;
    (S.equal) { }
      S.slice b0 src (src + i) `S.append` S.slice b0 (src + i) (src + i + 1);
    (S.equal) { lemma_slice_ijk b0 src (src + i) (src + i + 1) }
      S.slice b0 src (src + (i + 1));
    };
    let s1 = S.slice b1 (src + (i + 1)) (src + l) in
    let s0 = S.slice b0 (src + (i + 1)) (src + l) in
    let aux (j: nat { j < S.length s0 }): Lemma (S.index s0 j == S.index s1 j)
      [ SMTPat (S.index s0 j); SMTPat (S.index s1 j) ]
    =
      calc (==) {
        S.index s0 j;
      (==) {}
        S.index (S.slice b0 (src + i) (src + l)) (j + 1);
      (==) {}
        S.index (S.slice b1 (src + i) (src + l)) (j + 1);
      (==) {}
        S.index s1 j;
      }
    in
    ()
  in
  C.Loops.for 0ul l inv f


inline_for_extraction
val shrink:
  #a:Type -> vec:vector a ->
  new_size:uint32_t{new_size <= size_of vec} ->
  HST.ST (vector a)
  (requires (fun h0 -> live h0 vec /\ freeable vec))
  (ensures (fun h0 r h1 ->
                live h1 vec /\ live h1 r /\ size_of r = new_size /\
                frameOf r = frameOf vec /\
                hmap_dom_eq h0 h1 /\
                freeable r /\
                modifies (loc_vector vec) h0 h1 /\
                loc_vector vec == loc_vector r /\
                S.equal (S.slice (V.as_seq h0 vec) 0 (U32.v new_size))
                        (S.slice (V.as_seq h1 r) 0 (U32.v new_size))))
let shrink #a vec new_size =
  Vec new_size (Vec?.cap vec) (Vec?.vs vec)


inline_for_extraction
val flush_inplace:
  #a:Type -> vec:vector a -> 
  i:uint32_t{i <= size_of vec} ->
  HST.ST (vector a)
    (requires (fun h0 ->
      live h0 vec /\ freeable vec /\
      HST.is_eternal_region (frameOf vec)))
    (ensures (fun h0 fvec h1 ->
      frameOf vec = frameOf fvec /\
      hmap_dom_eq h0 h1 /\
      live h1 fvec /\ freeable fvec /\
      modifies (loc_vector vec) h0 h1 /\
      loc_vector vec == loc_vector fvec /\
      size_of fvec = size_of vec - i /\
      S.equal (V.as_seq h1 fvec)
              (S.slice (V.as_seq h0 vec) (U32.v i) (U32.v (size_of vec)))))
let flush_inplace #a vec i =
  let h0 = HST.get() in
  if i >= size_of vec then
    shrink vec 0ul
  else if i = 0ul then
    vec
  else begin
    let n_shifted = size_of vec - i in
    move_left (Vec?.vs vec) 0ul i n_shifted;
    shrink vec n_shifted
  end


inline_for_extraction
val rv_flush_inplace: 
  #a:Type0 -> #rst:Type -> #rg:regional rst a ->
  rv:rvector rg -> i:uint32_t{i <= size_of rv} ->
  HST.ST (rvector rg)
    (requires (fun h0 -> rv_inv h0 rv))
    (ensures (fun h0 frv h1 ->
      V.size_of frv = V.size_of rv - i /\
      V.frameOf rv = V.frameOf frv /\
      modifies (loc_rvector rv) h0 h1 /\
      rv_inv h1 frv /\
      S.equal (as_seq h1 frv)
              (S.slice (as_seq h0 rv) (U32.v i) (U32.v (V.size_of rv)))))
let rv_flush_inplace #a #rst #rg rv i =
  let hh0 = HST.get () in
  (if i = 0ul then () else free_elems rv (i - 1ul));
  rv_loc_elems_included hh0 rv 0ul i;

  let hh1 = HST.get () in
  assert (modifies (rs_loc_elems rg (V.as_seq hh0 rv) 0 (U32.v i)) hh0 hh1);
  let frv = flush_inplace rv i in

  let hh2 = HST.get () in
  assert (modifies (loc_region_only false (V.frameOf rv)) hh1 hh2);

  // Safety
  rs_loc_elems_disj
    rg (V.as_seq hh0 rv) (V.frameOf rv) 0 (U32.v (V.size_of rv))
    0 (U32.v i) (U32.v i) (U32.v (V.size_of rv));
  rs_loc_elems_parent_disj
    rg (V.as_seq hh0 rv) (V.frameOf rv)
    (U32.v i) (U32.v (V.size_of rv));
  rs_elems_inv_preserved
    rg (V.as_seq hh0 rv) (U32.v i) (U32.v (V.size_of rv))
    (loc_union (rs_loc_elems rg (V.as_seq hh0 rv) 0 (U32.v i))
               (loc_region_only false (V.frameOf rv)))
    hh0 hh2;
  assert (rv_inv #a #rst #rg hh2 frv);

  // Correctness
  as_seq_seq_preserved
    rg (V.as_seq hh0 rv) (U32.v i) (U32.v (V.size_of rv))
    (loc_union (rs_loc_elems rg (V.as_seq hh0 rv) 0 (U32.v i))
               (loc_region_only false (V.frameOf rv)))
    hh0 hh2;
  as_seq_seq_slice
    rg hh0 (V.as_seq hh0 rv) 0 (U32.v (V.size_of rv))
    (U32.v i) (U32.v (V.size_of rv));
  assert (S.equal (S.slice (as_seq hh0 rv) (U32.v i) (U32.v (V.size_of rv)))
                  (as_seq_seq rg hh2 (V.as_seq hh0 rv)
                    (U32.v i) (U32.v (V.size_of rv))));
  as_seq_seq_eq
    rg hh2 (V.as_seq hh0 rv) (V.as_seq hh2 frv)
    (U32.v i) (U32.v (V.size_of rv)) 0 (U32.v (V.size_of frv));
  assert (S.equal (as_seq_seq rg hh2 (V.as_seq hh2 frv)
                    0 (U32.v (V.size_of frv)))
                  (as_seq_seq rg hh2 (V.as_seq hh0 rv)
                    (U32.v i) (U32.v (V.size_of rv))));
  assert (S.equal (S.slice (as_seq hh0 rv) (U32.v i) (U32.v (V.size_of rv)))
                  (as_seq hh2 frv));
  frv