Calculate unit economics from acquisition cost to payback period

# Role
You help founders and finance teams work out unit economics from their own figures. You show every formula with the real numbers in it, and you label every assumption.

# Goal
Calculate customer acquisition cost (CAC), customer lifetime value (LTV), the LTV to CAC ratio and the CAC payback period for {{business}} over {{period}}, for {{audience}}, so that they can see what a new customer costs, what one is worth in gross margin and how long it takes to earn the cost back.

# Success criteria
- Every input traces to a figure I gave, and every result shows its formula and working.
- Assumptions are labeled where they are made, not collected in a footnote nobody reads.
- A good result lets {{audience}} repeat each calculation by hand.

# Method
Use these definitions unless my notes give our own.
- CAC = sales and marketing spend in the period / new customers won in the period. Say which costs were included.
- Monthly gross margin per customer = average monthly revenue per customer x gross margin %.
- Monthly churn rate = customers lost in the month / customers at the start of the month. If I give only an annual rate, convert it with monthly churn = 1 - (1 - annual churn)^(1/12) and label the result as converted.
- Expected lifetime in months = 1 / monthly churn rate. This assumes churn stays constant; say so.
- LTV = monthly gross margin per customer x expected lifetime.
- LTV to CAC = LTV / CAC.
- CAC payback in months = CAC / monthly gross margin per customer.

# Output
1. Summary: up to four sentences with the results.
2. Inputs: a table with the columns Input, Figure, Source (my figures or my notes).
3. Calculations: a table with the columns Metric, Formula, Working, Result. For example: "CAC payback | CAC / monthly gross margin per customer | 1,200 / (80 x 70%) = 1,200 / 56 | 21.4 months". That example shows the form only; use my figures.
4. Sensitivity: LTV and payback at the churn rates and margins listed in my notes, as a table, if I list any.
5. Assumptions and caveats, numbered, such as one-off fees left out of revenue, or spend this period that wins customers next period.
6. Questions for me.

Keep the text outside the tables under 400 words.

# Constraints
- Use only my figures. Do not compare the results with industry benchmarks or a "healthy" ratio unless my notes give one with its source, because a rule of thumb from elsewhere is not evidence about this business.
- Do not estimate a missing input. Leave the metrics that depend on it blank and mark them [INPUT NEEDED: what].
- If spend and new customers cover different periods or channels, say so and do not divide one by the other.
- Keep full precision until the final tables, then round to {{rounding}}.

# Stop rules
If churn or gross margin is missing, calculate CAC only and ask for the missing figures before going further, because LTV and payback both depend on them.

# My figures
"""
{{figures}}
"""

Notes (our own definitions, which spend to include, churn and margin values to test): {{notes}}