# Input Section (Referred to as T)
INPUT_TEXT = “[Insert the text to potentially transform]”
# Objective
Given INPUT_TEXT (T) above, determine if it benefits from a mathematical or symbolic representation. If so, rewrite it accordingly; otherwise, leave it unchanged. The system will attempt to formalize it mathematically if it leads to improved clarity and consistency; if not, it returns the original text.
# Step-by-Step Instructions
1. Analyze INPUT_TEXT (T)
* Check if representing the content with mathematical notation would make the instructions or concepts clearer and more logically consistent
* Ask yourself:
* Would expressing parts of T using equations, functions, sets, or logical symbols provide a clearer framework?
* Would doing so eliminate ambiguities or inconsistencies present in the original text?
3. Decision Criterion
* If “Yes” to both clarity and consistency improvements:
* Proceed with transformation
* If “No”:
* State: “No mathematical transformation is needed”
* Return the original text T as-is
5. Transformation (If Applicable)
* Identify and define sets, parameters, variables, and constants referenced in T
* Convert relationships, steps, rules, or processes into appropriate mathematical forms:
* Use equalities (x = y), inequalities (x ≤ y), or logical symbols (∧, ∨, ¬) as needed
* If multiple cases are described, consider piecewise functions
* Preserve meaningful textual context that does not benefit from a symbolic form
7. Output Guidelines
* If transformed: Provide a neatly formatted mathematical representation integrated with minimal explanatory text
* If not transformable: Explicitly state that no transformation is needed and return T unchanged
# Example Scenarios
## Example 1: Temperature Conversion
INPUT_TEXT (T): “Convert temperature from Celsius to Fahrenheit. For water, freezing is at 0°C and boiling is at 100°C.”
Transformation (Beneficial):
```
Let C = temperature in Celsius
Let F = temperature in Fahrenheit
F = 1.8C + 32
Notable points:
- Freezing point: C = 0 → F = 32
- Boiling point: C = 100 → F = 212
```
## Example 2: Profit Calculation
INPUT_TEXT (T): “Calculate profit where profit equals revenue minus costs. If costs exceed revenue, there is a loss. Revenue is sales times price per unit.”
Transformation (Beneficial):
```
Let:
R = revenue
C = costs
P = profit
S = number of sales
U = price per unit
R = S × U
P = R - C
Where:
P > 0: profit
P < 0: loss
P = 0: break-even
```
## Example 3: Game Rules
INPUT_TEXT (T): “In this card game, you win if you have all four aces or if you have three kings and at least two queens. You lose if you have no face cards.”
Transformation (Beneficial):
```
Let:
A = number of aces
K = number of kings
Q = number of queens
F = number of face cards
Win condition: (A = 4) ∨ (K = 3 ∧ Q ≥ 2)
Lose condition: F = 0
```
## Example 4: Story Request
INPUT_TEXT (T): “Write a creative story about a dragon who learns to bake cookies.”
Transformation (Not Beneficial): “No mathematical transformation is needed for this request.” [Return original text T]
## Example 5: Schedule Logic
INPUT_TEXT (T): “The shop is open Monday through Friday from 9 AM to 5 PM, except on holidays. During weekends, it’s open from 10 AM to 3 PM.”
Transformation (Beneficial):
```
Let t = time of day (in 24-hour format)
Let d = day of week
Let H = set of holidays
Opening hours = {
if d ∈ {Mon, Tue, Wed, Thu, Fri} ∧ d ∉ H:
9 ≤ t ≤ 17
if d ∈ {Sat, Sun}:
10 ≤ t ≤ 15
if d ∈ H:
closed
}
```
## Example 6: Recipe Instructions
INPUT_TEXT (T): “To make the cake, mix 2 cups flour with 1 cup sugar, add 3 eggs, and bake at 350°F for 30 minutes.”
Transformation (Not Beneficial): “No mathematical transformation is needed for this request.” [Return original text T]
## Example 7: Population Growth
INPUT_TEXT (T): “The population grows by 2% each year, starting from 1000 people.”
Transformation (Beneficial):
```
Let:
P(t) = population at year t
P(0) = 1000 (initial population)
r = 0.02 (growth rate)
P(t) = P(0)(1 + r)^t
```
## Example 8: Customer Acquisition Cost (CAC)
INPUT_TEXT (T): “Calculate the customer acquisition cost by taking total sales and marketing expenses for the quarter and dividing by the number of new customers acquired in that quarter.”
Transformation (Beneficial):
```
Let:
CAC = Customer Acquisition Cost
E = Total sales and marketing expenses
N = Number of new customers
t = Time period (quarter)
CAC(t) = E(t) / N(t)
Efficiency Criterion:
CAC(t) < CAC(t-1) indicates improving efficiency
```
## Example 9: SaaS Revenue Forecasting
INPUT_TEXT (T): “Project monthly recurring revenue assuming 5% customer churn rate, 15% expansion revenue from existing customers, and 20 new customers per month at $100 per customer.”
Transformation (Beneficial):
```
Let:
MRR(t) = Monthly Recurring Revenue at month t
c = 0.05 (churn rate)
e = 0.15 (expansion rate)
n = 20 (new customers per month)
p = 100 (price per customer)
MRR(t) = MRR(t-1)(1 - c)(1 + e) + n×p
```
## Example 10: Sales Pipeline Conversion
INPUT_TEXT (T): “Track conversion rates through the sales pipeline: 1000 leads → 300 qualified leads → 100 opportunities → 20 closed deals. Calculate conversion rate at each stage.”
Transformation (Beneficial):
```
Let:
L = Initial leads = 1000
QL = Qualified leads = 300
O = Opportunities = 100
D = Closed deals = 20
Conversion rates:
Lead → Qualified: QL/L = 30%
Qualified → Opportunity: O/QL = 33.3%
Opportunity → Deal: D/O = 20%
Overall conversion: D/L = 2%
```
## Example 11: Product-Market Fit Survey
INPUT_TEXT (T): “Analyze product-market fit survey where users rate how disappointed they would be if the product disappeared: 40% very disappointed, 35% somewhat disappointed, 25% not disappointed.”
Transformation (Beneficial):
```
Let:
V = Very disappointed = 0.40
S = Somewhat disappointed = 0.35
N = Not disappointed = 0.25
Product-Market Fit Score = V × 100
Decision criteria:
PMF = {
if V ≥ 0.40: Strong product-market fit
if 0.25 ≤ V < 0.40: Moderate fit
if V < 0.25: Poor fit
}
```
## Example 12: Marketing Channel ROI
INPUT_TEXT (T): “Calculate ROI for each marketing channel: Paid ads spent $5000 generated 50 customers, Content marketing spent $3000 generated 30 customers, Sales team spent $10000 generated 40 customers. Each customer has $1000 lifetime value.”
Transformation (Beneficial):
```
Let:
C = Customer lifetime value = $1000
Channels = {Paid, Content, Sales}
For each channel i ∈ Channels:
ROI(i) = (Revenue - Cost)/Cost × 100
where:
Revenue = Customers(i) × C
Results:
Paid: ROI = (50×1000 - 5000)/5000 × 100 = 900%
Content: ROI = (30×1000 - 3000)/3000 × 100 = 900%
Sales: ROI = (40×1000 - 10000)/10000 × 100 = 300%
```
## Example 13: Freemium Conversion Strategy
INPUT_TEXT (T): “With freemium model, target 5% conversion rate from free to paid tier within 3 months of signup, with premium features unlocking after 10 uses or when storage exceeds 1GB.”
Transformation (Beneficial):
```
Let:
U = Number of feature uses
S = Storage used (GB)
t = Time since signup (months)
CR = Conversion rate
Target: CR ≥ 0.05 when t ≤ 3
Upgrade triggers = {
if U > 10: Trigger premium
if S > 1: Trigger premium
if t > 3 ∧ CR < 0.05: Review strategy
}
```
## Example 14: Net Revenue Retention
INPUT_TEXT (T): “Calculate net revenue retention by taking current revenue from existing customers divided by revenue from same customers 12 months ago, including expansions, contractions, and churn.”
Transformation (Beneficial):
```
Let:
R₀ = Revenue from cohort at start
R₁ = Current revenue from same cohort
E = Expansion revenue
C = Contraction revenue
L = Lost revenue (churn)
NRR = R₁/R₀ × 100
where:
R₁ = R₀ + E - C - L
Healthy SaaS criteria:
NRR > 100%: Growing revenue per customer
NRR = 100%: Stable revenue
NRR < 100%: Shrinking revenue
```
## Example 15: A/B Test Targeting
INPUT_TEXT (T): “For the email campaign, show version A to enterprise customers with over 100 users or high usage, and version B to everyone else.”
Transformation (Beneficial):
```
Let:
U = Number of users
H = High usage flag (boolean)
E = Enterprise customer flag (boolean)
Version assignment = {
if (U > 100 ∨ H = true) ∧ E = true: Show version A
else: Show version B
}
```
# Guidelines
## Transformation Criteria
1. Transform text when it contains:
* ✓ Mathematical relationships (e.g., revenue = price × quantity)
* ✓ Logical conditions (e.g., if X then Y)
* ✓ Quantitative calculations
* ✓ Time-based rules (e.g., scheduling logic)
* ✓ State transitions (e.g., status changes)
* ✓ Optimization problems (e.g., maximizing ROI)
2. Do NOT transform text that is:
* ✗ Narrative or creative writing
* ✗ Simple instructions (step-by-step guides)
* ✗ Descriptive content
* ✗ Subjective discussions or opinions
* ✗ General advice
* ✗ Emotional or personal content
## Transformation Rules
1. Simplicity & Clarity
* Keep transformations as simple as possible
* Use mathematical notation only when it improves understanding
* Keep variables and symbols intuitive (e.g., ‘r’ for rate)
2. Format & Notation
* Use standard mathematical notation when available
* Format output for clear readability
* Start with variable definitions (“Let x = ...“)
* Group related equations together
* Include units where relevant
3. Context & Explanation
* Preserve important contextual text
* Include brief explanations when variables aren’t self-evident
* Add explanatory notes where needed for clarity
* Keep crucial context in plain text
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