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:

PLY parser - equation assignation

I'm creating a calculator using PLY. 
I want to be able to assign equation to a dictionary as I am able to assign variables to another dictionary.
The way I'm assigning variables : x = 10 (in the dict xwill be the key and 10the value)
The way I want to be able to assign equation : fun(x) = x + 42 (in the dict funwill be the key and a tuple ('x', 'x+10') will be the value).
It's working with this writing fun|x| = x + 42 (note the 'pipe' symbol here). But it's not working with parentheses. 
How can I make it work the right way ? 
Here's my code so far : 
import ply.yacc as yacc
import ply.lex as lex

################################## LEXER ################################

tokens = (
    'NAME',
    'NUMBER',
)

t_NAME      = r'[a-zA-Z]+'
literals    = '+=-*/|()'
t_ignore    = " \t"

def t_NUMBER(t):
    r'(?:\d+(?:\.\d*)?)'
    t.value = int(t.value)
    return t

def t_error(t):
    print('Illegal character \'{}\''.format(t.value[0]))
    t.lexer.skip(1)

################################## PARSER ################################

functions = {}
variables = {}

def p_operations(p):
    """ statement : expression
    expression : fifth
    fifth : fourth
    fourth : third
    third : second
    second : first
    first : NUMBER
    """
    p[0] = p[1]

def p_plus(p):
    """ fifth : fifth '+' fourth """
    if isinstance(p[1], str) or isinstance(p[3], str):
        p[0] = str(p[1]) + p[2] + str(p[3])
    else:
        p[0] = p[1] + p[3]

def p_minus(p):
    """ fifth : fifth '-' fourth """
    if isinstance(p[1], str) or isinstance(p[3], str):
        p[0] = str(p[1]) + p[2] + str(p[3])
    else:
        p[0] = p[1] - p[3]

def p_implicit_times(p):
    """ fourth : fourth second """
    if isinstance(p[1], str) or isinstance(p[2], str):
        p[0] = str(p[1]) + str(p[2])
    else:
        p[0] = p[1] * p[2]

def p_times(p):
    """ fourth : fourth '*' third """
    if isinstance(p[1], str) or isinstance(p[3], str):
        p[0] = str(p[1]) + p[2] + str(p[3])
    else:
        p[0] = p[1] * p[3]

def p_divide(p):
    """ fourth : fourth '/' third """
    if isinstance(p[1], str) or isinstance(p[3], str):
        p[0] = str(p[1]) + p[2] + str(p[3])
    else:
        p[0] = p[1] / p[3]

def p_unary_minus(p):
    """ third : '-' third """
    if isinstance(p[2], str):
        p[0] = '-' + p[2]
    else:
        p[0] = -p[2]

def p_power(p):
    """ second : first '^' third """
    if isinstance(p[1], str) or isinstance(p[3], str):
        p[0] = str(p[1]) + p[2] + str(p[3])
    else:
        p[0] = p[1] ** p[3]

def p_block(p):
    """ first : '(' expression ')' """
    p[0] = p[2]

################################ PROBLEM HERE ############################

def p_statement_assign(p):
    ''' statement : NAME '=' expression '''
    variables[p[1]] = p[3]
    p[0] = p[3]

def p_function_assign(p):
    ''' statement : NAME '|' expression '|' '=' expression '''
    functions[p[1]] = (p[3], p[6])
    p[0] = functions[p[1]]

def p_variable_expr(p):
    ''' first : NAME '''
    try : 
        p[0] = variables[p[1]]
    except:
        p[0] = p[1]

def p_error(t):
    print("Syntax error!")

################################## MAIN #################################

lexer = lex.lex() 
parser = yacc.yacc()

while True:
    try:
        question = raw_input('>>> ')
    except:
        question = input('>>> ')

    answer = parser.parse(question)
    if answer is not None:
        print(answer)

A:

You allow implicit multiplication. That means that f(x) can be parsed as a product, in which case f would have to be reduced to fourth as per the implicit multiplication rule. But if it is to be parsed as an assignment, it needs to be left as a NAME. That's a shift-reduce conflict, which can easily be seen in parser.out:
state 3

    (16) statement -> NAME . = expression
    (17) statement -> NAME . ( expression ) = expression
    (18) first -> NAME .

  ! shift/reduce conflict for ( resolved as shift

What you see here is that when the parser sees NAME followed by (, it doesn't know whether to reduce NAME to first (and subsequently to second, etc. up to fourth), in the expectation that the statement is a simple computation, or to shift the (, thereby committing itself to treating this as a function definition.
You might already have experienced a similar problem, since a natural grammar for function definition is:
statement : NAME '(' NAME ')' '=' expression

but you've replaced the second NAME with expression. That will avoid the shift-reduce conflict before the ), at the cost of accepting questionable function definitions (f(a+3) = 2a).
It's possible to do something similar to avoid this shift-reduce conflict (but it's a very ad hoc solution):
statement : fourth '(' expression ')' '=' expression

That "works" in the sense that it accepts correct expressions. But it also silently (or somewhat silently) accepts quite a few other expressions:
This one is fine:
>>> f(a) = a + 3
('a', 'a+3')

But these are weird:
>>> -f(a) = a + 3
('a', 'a+3')
>>> 3f(a) = a + 3
('a', 'a+3')
>>> 3f(a+2) = a + 3
('a+2', 'a+3')

Alternatively, you could ignore the shift-reduce conflict, since PLY will, by default, do the shift (as it says in parser.out: "resolved as shift"). That will prevent the parser from accepting the weird examples above, but it will also incorrectly parse some expressions which might seem reasonable:
These ones seem appropriate:
>>> f(a) = a + 3
('a', 'a+3')
>>> -f(a) = a + 3
Syntax error!
a+3
>>> 3f(a) = a + 3
Syntax error!
a+3

But we might expect this to print 105:
>>> a=7
7
>>> a(2a+1)
Syntax error!

If that doesn't bother you, you can stop reading now.

Your grammar is not ambiguous, and nor would it be ambiguous if you'd written the definition syntax with greater specificity:
statement : NAME '(' NAME ')' '=' expression

or, if you wanted to allow functions with multiple arguments:
statement : NAME '(' name_list ')' '=' expression
name_list : NAME
name_list : name_list ',' NAME

But it is not LR(1), and it will be very difficult to make it LR(1). Both of the syntaxes suggested above are LR(4), and since every LR(k) grammar can theoretically be converted mechanically into a (very bloated and hardly readable) LR(1) grammar, a solution must exist. It won't be pretty, though.
(The actual syntax you are using, with expression, is not LR(k) for any k, because expression can be arbitrarily long and the parser has to see beyond the argument list in order to decide whether or not to reduce the first NAME.)
Since the grammar is unambiguous, you could parse it with a GLR/GLL/Earley/etc. parser, but PLY doesn't produce those and I don't know of a Python parser generator which does (although that doesn't mean that one doesn't exist). There are various GLR parser generators available for other languages.
With PLY, though, your best bet is probably to use the general syntax shown above as an ad hoc solution and then solve the problem of accepting incorrect definitions by doing a check in the semantic action. 
However, that check will be a bit tricky unless you bite the bullet and change to a parser which produces ASTs instead of doing immediate evaluation, as we have discussed several times. If the semantic values are ASTs, then it will be straight-forward to verify that both p[1] and p[3] are simple names in the action function for 
statement : fourth '(' expression ')' '=' expression

I suppose, human ingenuity being limitless, that you might find some other hack which lets you do the test. But don't expect much sympathy when it fails on a corner case.