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]


Q:

Suslin representation of sets and limits to Shoenfield's Absoluteness

For any natural number $k \in \omega$ and any set $X$ the set $$T \subseteq \bigcup_{m \in \omega} (\omega^m)^k \times X^m $$ is a tree on $\omega^k \times X$ iff 
$$(t_o, \ldots, t_k) \in T \: \Rightarrow \forall m \leq |t_0| \: (t_0 \upharpoonright m, \ldots, t_k \upharpoonright m) \in T,$$
i.e. the elements of $T$ are $k+1$-tuples of finite sequences of the same length.
It is known that any tree $T$ on $\omega^k \times X$ is well-founded if there is an order-preserving function $\rho: T \rightarrow Ord$.
Let $x = (x_0, x_1, \ldots x_{l-1}) \in (\omega^\omega)^l$. The set $T(x)$ is a tree on $\omega^{k-l} \times X$ defined as follows:
$$(t_0, \ldots, t_{k-l}) \in T(x) \: \Leftrightarrow \: \exists m \: (x_0 \upharpoonright m, \ldots, x_{l-1} \upharpoonright m, t_0, \ldots, t_{k-l}) \in T.$$
It is also well-known that if $a \in \omega^\omega$ and $A \subseteq (\omega^\omega)^k$, then $A$ is $\Pi^1_1(a)$ iff there is a tree $T$ on $\omega^k \times \omega$ such that for any real $x \in (\omega^\omega)^k$:
$$x \in A \: \Leftrightarrow \: T(x) \text{  is well-founded,}$$
where the relation $\{(i, x): s_i \in T(x)\}$ is computable from $a$, for some computable enumeration $(s_i)_{i \in \omega}$ of the set of finite sequences such that $\forall i \: |s_i| \leq i$.
Now the crucial definition: for a set $A \subseteq (\omega^\omega)^k$ and any set $X$ we say that $A$ is $X$-Suslin if there i a tree $T$ on $\omega^k \times X$  such that$A = p[T]$, i.e. for any $x \in (\omega^\omega)^k$ it holds that
$$ x \in A \: \Leftrightarrow \: T(x) \text{  is ill-founded}.$$
To illustrate the definition, note that every analytic set is $\omega$-Suslin, and observe that (under CH?) any set $A \subseteq (\omega^\omega)^k$ is $2^{\aleph_0}$-Suslin, but an important thing is that the key lemma in the (standard) proof of Shoenfield's Absolutness is the fact that 
 any $\mathbf{\Sigma^1_2}$ set is $\omega_1$-Suslin (a remark from ref. Kanamori - The Higher Infinite p.174: the reverse implication is independent of ZFC). The way you prove the key lemma is you take an appropriate well-founded tree of triples (by the definition of $\mathbf{\Sigma^1_2}$), using the above-mentioned fact that there is an order-preserving map $\rho$ from $T$ to $\omega_1$ and you define a new tree $T(x)$ on $\omega^2 \times \omega_1$, ewentually transforming it into a tree with a branch that gives us the representation - if necessary, I can put in more details into the question.
My question is as follows (I guess the answer is in the negative, but cannot find an argument): 
$$\text{is it true that every $\mathbf{\Sigma^1_{n+1}}$ set is $\omega_n$-Suslin?}$$
If not, what step from the proof of the case of $n=1$ does not generalize? Is it simply (that's my intuition) the fact that when we go higher in the projective hierarchy, we simply lose the ability to use the definition of our set to form an appropriate tree in the first place (in the sense in which having a $\Sigma^1_2(a)$ predicate gives us a formula 'there exists $r$ such that for all s some $\Pi^0_1$ formula holds' and this easily translates into a tree that we can handle quite feasibly)? If so, is there a proper formal argument for that?
I am obviously also interested in any counterexample, but the most important thing for me is to understand which steps of the argument brake if they do?
The reason for the question is the Shoenfield's Absolutness. We can give nice counterexamples of $\mathbf{\Sigma^1_3}$ formulas (like 'there is a non-constructible real') that are not absolute for (transitive) inner models, so we know $\mathbf{\Sigma^1_2}$ (and $\mathbf{\Pi^1_2}$) is the limit, but when asked 'which step of the proof cannot be generalized?', my reply can be only: well, I know how to handle $\Pi^1_1(a)$ properties (and quantify them existentially to handle $\Sigma^1_2(a)$-ones), but I don't see a way what I could do for some $\Sigma^1_{n}(a)$ for $n>2$. Is it just this or there is some deeper, say, 'combinatorial' reason within some of the further steps of the proof that prevents us from the generalization? 

A:

I don't know the answer in ZFC, but in ZF + DC the answer is consistently no.  If $V = L$ and there is an inaccessible cardinal, then in the associated Solovay model the $\Pi^1_2$ relation $\{ \langle x,y \rangle : y \notin L[x]\}$ has no uniformization, so it is not Suslin, meaning that it is not $X$-Suslin for any wellordered set $X$.
(Moreover, in the Solovay model of a slightly stronger large cardinal, every Suslin set of reals is ${\bf\Sigma}^1_2$ by Theorem 1.8 and Remark 1.9 of Wilson, Generic Vopěnka cardinals and models of ZF with few $\aleph_1$-Suslin sets: https://link.springer.com/article/10.1007/s00153-019-00662-1.)
Consequently, the answer to a version of the question for definable Suslin representations in ZFC is also no: if $V = L$ and there is an inaccessible cardinal, then in the generic extension by its Levy collapse there is a $\Pi^1_2$ set with no Suslin representation definable from a countable sequence of ordinals.
To go from $\omega$-Suslin representations of $\Sigma^1_1$ sets to $\omega_1$-Suslin representations of $\Pi^1_1$ sets, we use the fact that if $T$ is a countable tree, then rank functions witnessing well-foundedness of its sections $T(x)$ can be coded as $\omega$-sequences of countable ordinals and therefore as branches of order type $\omega$ through a tree of cardinality $\omega_1$.  The most straightforward attempt to imitate this argument at the next level would seem to involve branches of order type $\omega_1$ through a tree of cardinality $\omega_2$.  But the kind of tree that we are talking about can only have branches of order type $\le \omega$ by definition (and this restriction is necessary for such trees to be relevant to uniformization, etc. of sets of reals.)
It should be mentioned here that if projective determinacy holds, then all projective sets have definable Suslin representations by the second periodicity theorem of Moschovakis.