#BeyondVSM: Understanding and Mapping Your Process of Knowledge Discovery

This short post will serve as the “table of contents” for a series of six posts I wrote this year about mapping processes in creative industries based on knowledge creation and information discovery and not by handoffs of work between people and teams.

  • Understanding Your Process as Collaborative Knowledge Discovery: the first post in the series explores the problem and the new ways of looking at it
  • Examples of using this approach to map processes of knowledge discovery in two different industries
  • Mapping Your Process as Collaborative Knowledge Discovery. I wrote about how to actually create such process maps with real people who do the work, why, and how to use these maps in Kanban system design. This post turned into three, each covering different layers:
    • Recipes: how to actually do it and what not to do, without much explaining why. Sorry, the post was already long enough. Of course, recipes are not enough, but that’s what I have the next two posts for!
    • Observations: what actually happened as I tried this new approach. The experience informed various tips on how to do it.
    • Thinking
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Lead Time and Iterative Software Development

I have introduced my forecasting cards and written about lead time distributions in my recent blog post series. Now I’d like to turn to how these concepts apply in iterative software development, particularly the popular process framework Scrum.

Let’s consider one of the reference distribution shapes (Weibull k=1.5), which often occurs in product development, particularly software. I went through various points on this curve and replaced them with what they ought to mean in this specific context.

Lead time distributions and the timebox. The chart shows the mode, the median, the average, and the 75th percentile relative to the sprint duration

Scrum teams often complain that their user stories are not finished in the same sprint they were started. I have often observed in such situations that their stories are simply too large.

Even if typical stories were smaller than the duration of the sprint, such as, 7-8 days in a 10-business-day, two-week sprint, that was not small enough. The teams, Scrum masters, product owners held, perhaps subconsciously, the notion that we can “keep the average and squeeze the variance”, that is keep the 7-8-day average but limit variability — estimate, plan and task better — so that the right side of the distribution fits within the timebox. Recent lead time distribution research, examining many data sets from different companies (including those using iterative Agile methods) refute this notion. One of the key properties of common lead time distributions is that the average and standard deviation are not independent.

Another suggestion — keeping the average story to half the sprint duration, so that the ends of the bell curve gives us zero in the best case and the sprint duration in the worst case — is another illusion. Lead time distributions are asymmetric!

Leftshifting diagram: as the lead time distribution curve shifts to the left, very few data points don't fit into the timebox

The real strategy is to left-shift the whole distribution curve.

This Kanban-sourced knowledge led to many quick wins as the Scrum teams, their Scrum masters and product owners I coached gave themselves a goal to systematically make their stories smaller. They simply asked, what can we do to double the count of delivered stories in the next few sprints, covering roughly the same workload in each sprint? After the doubling, ask the same question again until the stories are small enough.

How Small?

How small do user stories need to be? We can turn to our forecasting cards, which give the control-limit-to-average ratios between 3.9 and 4.9 for the two most common distribution shapes (1.25 and 1.5). In the extreme case, we have to assume the exponential distribution (I have observed quasi-exponential distributions in some cases in incremental software development), which gives us the ratio of 6.6. The ratio of average lead time to sprint duration in the range of 1:4 to 1:6 can be used as a guideline.

To make this rule of thumb a bit more practical, lets take into account these practical considerations: (1) the lead time is likely to be measured in whole days, (2) the number of business days in a sprint is likely to be a multiple of five, and (3) the median (half-longer, half-shorter) is easier to use in feedback loops than the average.

The control-limit-to-median ratios for the same distribution shapes are (consulting the forecasting cards again) 4.5 to 6.1; in the extreme case, 9.5. Therefore, half of the stories in one-fifth of the sprint duration can be used as a guideline. In the extreme cases, we may need one-tenth instead of one-fifth.

None of this is news to experienced Scrum practitioners, particularly those with eXtreme Programming backgrounds. XP tribe has appreciated the value of small stories since long ago, and invented and evangelized techniques, such as Product Sashimi to make them smaller.

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Introducing Lead Time Forecasting Cards

I’m introducing a simple tool: lead time forecasting cards.

The set of six (so far) forecasting cards

Each card displays a pre-calculated distribution shape, using Weibull distribution with shape parameters 0.75, 1 (Exponential distribution), 1.25, 1.5, 2 (Rayleigh distribution), and 3. (Since I printed the first batch, I realized I need to include k=0.5 in the collection.)

For each distribution, the following points are marked with rainbow colours:

  • mode
  • median
  • average
  • percentiles (63rd, 75th, 80th, 85th, 90th, 95th, 98th, and 99th)
  • the upper control limit (99.865%)

The scale of each card is such that the lead time average is 1. Your average is different, so multiply it by the numbers given in the table on each card.

I will be bringing a small number of printed cards to the upcoming conferences, training classes and consulting clients. The goal is, of course, to get feedback, refine them, and then make them more widely available.

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Lead Time Distributions and Antifragility

This post continues the series about lead-time distribution and deals with risks involved in matching real-world lead-time data sets to known distributions and estimating the distribution parameters.

Convex option payoff curve. Losses are limited on the left, gains are unlimited on the right

One of the key ideas of Nassim Nicholas Taleb’s book Antifragile is the notion of convexity, demonstrated by this option payoff curve. The horizontal axis shows the range of outcomes, the vertical axis shows the corresponding payoff. With this particular option, the payoff is asymmetric. Our losses in case of negative outcomes on the left are limited to a small amount. But our gains on the right side (positive outcomes) are unlimited. Note that this is due to the payoff function’s convexity. If the function was only increasing as a straight line, both our losses and gains would be unlimited.

A concave payoff function would achieve the opposite effect: limited gains and unlimited losses.

An antifragile system exposes itself to opportunities where benefits are convex and harm is concave and avoids exposure to the opposite.

In the book’s Appendix, Taleb considers a model that relies on Gaussian (normal) distribution. Suppose the Gaussian bell curve is centered on 0 and the standard deviation (sigma) is 1.5. What is the probability of the rare event that the random variable will exceed 6? It’s a number, and a pretty small one, which anyone with a scientific calculator can calculate.

Gaussian distribution analysis: probability of a rare event as a function of sigma is a convex function.

Gaussian distribution analysis: probability of a rare event as a function of sigma is a convex function.

Right? Wrong. We don’t really know that sigma is 1.5. We simply calculated it from a set of numbers collected by observing some phenomenon. The real sigma may be a little bit more or a little bit less. How does that change the probability of our rare event? There is a chart in the Appendix, but I rechecked the calculations, and here it is — it’s a (very) convex function.

If we overestimate sigma a little bit, it’s really less than what we think it is, on the left side of the chart, we overestimate the probability of our rare event — a little bit. But if we underestimate sigma a little bit, we underestimate the probability of our rare event — a lot.

Convexity Effects in Lead Time Distributions

Weibull distribution analysis: probability of exceeding SLA as a function of parameter

Weibull distribution analysis: probability of exceeding SLA as a function of parameter

Weibull distribution analysis: probability of exceeding SLA as a function of scale parameter (shape parameter k=3).

Weibull distribution analysis: probability of exceeding SLA as a function of scale parameter (shape parameter k=3).

Let’s apply this convexity thinking to lead time distributions of service delivery in knowledge work. Weibull distributions with various parameters are often found in this domain. Let’s say we have a shape parameter k=1.5 and a service delivery expectation: 95% of deliveries within 30 days. If we are spot-on with our model, the probability to fail this expectation is exactly 5%. How sensitive is this probability to the shape and scale parameters?

With respect to the shape parameter, the probabilities to exceed the SLAs are all convex decreasing functions (I added the SLAs based on 98th and 99th percentiles to the chart). If we underestimate the shape parameter a bit, we overstate the risk a bit; if we overestimate it a bit, we understate the risk — a lot.

In other distribution shape types (k<1, k>2), it is the same story. The risk of underestimating-overestimating the shape parameter is asymmetric.

What about the scale parameter? It turns out there is less sensitivity with respect to it. The convexity effect (it pays to overestimate the scale parameter) is present for k>2 (as shown by the chart), it is weaker for 1<k<2, and the curves are essentially linear for k<1.


When analyzing lead time distributions of service delivery in creative industries, it is important not to overestimate the shape parameter. The under-overstatement of risk due to the shape parameter error is asymmetrical.

Matching a given lead-time data set to a distribution doesn’t have to be a complicated mathematical exercise. We should also not fool ourselves about the precision of this exercise, especially given our imperfect real-world data sets. Using several pre-calculated reference shapes should be sufficient for practical uses such as defining service level expectations, designing feedback loops and statistical process control. If we find our lead-time data set fits somewhere between two reference shapes, we should choose the smaller shape parameter.

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How to Match to Weibull Distribution without Excel

A bit more than one year ago, I wrote a short, but fairly technical post on how to do this without complicated statistical tools, only using something found in many modern offices: spreadsheet software such as Excel.

Here is the problem we are trying to solve. Weibull distribution occurs often in service delivery lead times in various creative industries. We need to match given lead time data sets and find the distribution parameters (shape and scale), which help in understanding risks, establishing service-level expectations and creating forecasting models.

My post and the spreadsheet containing the necessary formulas (just copy and paste your own data) are still valid. However, I would like to propose a simpler method with even lower barrier to entry. It is less precise, but still reasonably accurate and you can use it to think on your feet without more complicated tools in your hands.

3 Main Shapes And 2 Boundary Cases

The three main shapes of Weibull distribution curves can be told from one another by observing convexity and concavity of the left side and the middle of the curve. (On the right side, they are all convex.) The three cases correspond to the shape parameter being (1) less than 1, (2) between 1 and 2, (3) greater than 2. Exponential (k=1) and Rayleigh (k=2) distributions give the boundary cases, separating three broad classes.

Let’s take a look at the charts.

Weibull distribution curve with shape parameter 0.75. The curve is convex over the entire range when the shape parameter is less than 1.

Weibull distribution, shape parameter k=0.75

Weibull distribution curve for the shape parameter 1.5

Weibull distribution, shape parameter k=1.5

Weibull distribution with shape parameter 1.25

Weibull distribution with shape parameter k=1.25

When the shape parameter is less than 1, the distribution curve (probability density function) is convex over the entire range. This shape parameter range occurs often in IT operations, customer care and other fields with a lot of unplanned work. The lowest value of the shape parameter that I have observed is 0.56. I chose k=0.75 as the representative of this class. The main visual features of this distribution are, besides convexity: the right-shifted average (for example, for k=0.75, the average falls on the 68th percentile) and a wide spread of common-cause variation (for example, for k=0.75, the ratio of the 99th percentile to the median is about 15:1).

When the shape parameter is between 1 and 2, the curve is concave on the left side and through the peak and turns convex on the back slope. This shape parameter range occurs in product development environments. I have observed shape parameters close to 1, close to 2 and everything in between, but parameters between 1 and 1.5 more often than between 1.5 and 2. I chose k=1.5 and k=1.25 as two representatives of this class. The main visual features of this distribution is the asymmetric “hump” rising steeply on the left side and sloping gently on the right. The mode (the distribution peak) is left-shifted (as I wrote earlier, it falls on the 18th and 28th percentiles, respectively, for k=1.25 and k=1.5). The median is slightly, but appreciably less than the average (87% of the average for k=1.5). The common-cause variation spread is significant, but narrower than for k<1.

Weibull distribution with shape parameter k=3

Weibull distribution with shape parameter k=3

In software engineering, the adoption of Agile methods may lead to left-shifting of the shape parameter. This corresponds to the team's growing ability to deliver software solutions in smaller batches faster. This was another reason for including k=1.25 as a reference point.

Exponential distribution, also Weibull distribution with shape parameter k=1

Exponential distribution, also Weibull distribution with shape parameter k=1

When the shape parameter is greater than 2, the curve is convex on the left side, then turns concave as it goes up towards the peak, and turns convex again on the back slope. This distribution shape often occurs in phase-gated processes. I have only few observations of this type of distribution curve and the greatest value of the shape parameter I have observed so far is 3.22. I chose k=3 as the representative of this class. The main visual feature of this distribution is that a much more symmetric peak (although it is slightly asymmetric). Compared to k<2, the spread of common-cause variation is narrower relative to the average, however, processes with this type lead time distribution tend to have very long average lead times.

Rayleigh distribution, also Weibull distribution with shape parameter k=2

Rayleigh distribution, also Weibull distribution with shape parameter k=2

The Exponential distribution (k=1) provides the boundary between two classes of shapes (k<1 and 1<k<2). It is a very well-studied distribution thanks to all the research of the queuing theory and Markov chains. This distribution has a unique property: its hazard rate, which is the propensity to finish yet unfinished work, remains constant and independent of time. When the hazard rate decreases with time, we get distributions such as with k<1; when it increases slowly, we get distributions such as the ones with 1<k<2 shown above.

Similarly, Rayleigh distribution (k=2) provides the boundary between the k>2 and 1<2<k classes.

Matching Your Data Set to Weibull Distribution

The first step is to narrow down the shape parameter range by observing convexity-concavity on the left side and in the centre.

The second step is to compare your lead time distribution histogram and to the few reference shapes provided in this post and see if it is close enough to one of them or is somewhere between the two. Ratios of various percentiles to the average can help. This will pin the shape parameter within a fairly narrow range (e.g. 1.25 to 1.5). This is not very precise, but accurate enough for many practical applications, such as reasoning about service-level expectations, control limits and feedback loops via the median.

Math Check

Finally, there are two quantitative shortcuts to estimating Weibull shape and scale parameters.

A good approximation of the scale parameter (thanks again to Troy for pointing this out) is the 63rd percentile. Indeed, the value of the Weibull cumulative distribution function where the random variable (lead time) equals the scale parameter is independent of the shape parameter and is given by:

The value of the quantile (inverse cumulative distribution) function at this “magic point” is indeed the scale parameter as shown by:

The “magic point” has another important property, expressed by the following formula:

If the lead time data set is smooth enough to allow approximating the slope of the quantile function at the 63rd percentile (using, for example, finite differences), then this formula can be used to estimate the shape parameter.


  • Weibull distribution occurs often in lead time data sets in creative industries. Estimating distribution parameters (shape and scale) is desirable so that we can use them in various quantitative models.
  • The linear regression method is still preferable for finding the shape and scale parameters.
  • In situations when the tools are not accessible, the shape and scale parameter can be approximated by comparison with several reference shapes. It is important to differentiate between three shape parameter ranges (k<1, 1<k<2, k>2) by observing convexity and concavity on the left side and the centre of the distribution curve.
  • Relatively low precision (two-tenths for the shape parameters) is still sufficient for practical, quick thinking about the service delivery capability described by the lead time data set.
  • There are two math tricks based on the unique properties of Weibull distribution and the 63rd percentile that can be used to roughly estimate the distribution parameters.
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Inside a Lead Time Distribution

My earlier post discussed:

  • how to measure lead time
  • how to analyze lead distribution charts
  • how to get better understanding of lead time by breaking down multimodal (work item type mix) lead time data sets into unimodal (by work item type)
  • how to establish service-level expectations based on lead time data

As a reminder, the context is creative endeavours in knowledge-work industries.

Now let’s take a closer look at the structure of lead-time distributions and their common elements.

Lead time distribution histogram with the best-fit distribution curve and key points on the curve marked with different colours

Mode. The mode is the most probable or, in other words, the most often occurring number in the data set, or the location of the peak of the distribution curve. In lead time distributions I often observe in delivery processes in knowledge work industries, the mode is often left-shifted and tops the distribution’s asymmetric “hump.” The probability of lead time being less than the mode is small, 18% to 28% observed in distribution shapes common in product development.

The mode is also what people tend to remember well. When asked how long it take to deliver this type of solution, they give the answer their memory is trained on. The trouble is, their memory may be trained on the 18th percentile, so beware of the remaining 82%.

Median. The median is also known as the 50% percentile. If we sorted all numbers in our data set, this one would be right in the middle. Half of the deliveries take more time, half take less. Lead time distributions observed in creative industries are asymmetrical and the median is usually less than the average, sometimes significantly. The ratio of median-to-average of 80% to 90% is quite common in product development. It can drop to 50% in operations and customer care.

Because the median is located on the left side of many other important points on the distribution, it is very useful in establishing short feedback loops to continuously validate forecasting models and project and release plans. If half of the lower-level work items (such as, features in a product release, tasks in a project, and so on) are still being delivered in less than or equal time, in other words, the median is not drifting to the right, the original forecast is still sound.

This insight is due to David J. Anderson, Dan Vacanti and their staff at Corbis. This is a rarely publicized part of their well-known case study from 2006-07. While the project was large and took a long time to deliver in total, there was a feedback loop in it taking only a few days to run and closing almost every day and revalidating the project forecast. The understanding of the median and its role in the distribution enabled this feedback loop.

Average. The average is the easiest to calculate. It connects with the work-in-process (WIP) and throughput in a simple equation known as Little’s Law.

The 63rd percentile. In the Weibull family of distributions, which are often observed, there is one special point, where the over and under probabilities don’t depend on the shape parameter. This point is

The formula to calculate the "magic point": 1 minus 1 over e (where e=2.71828) equals approximately 0.63

Credit to Troy Magennis for pointing this out. Thanks to its unique properties, this percentile can be used in estimating the scale parameter when matching lead time data sets to Weibull distribution.

The 75th percentile. The percentiles between 50th and 75th can be used to estimate the shape parameter if we’re dealing with Weibull distribution and the shape is known to be between 1 and 2, which is common in product development. This owes to another unique property of the 63rd percentile. The math of this estimation is not the purpose of this post, so I’ll save it for future writing.

Higher percentiles (80th and up). The higher percentiles are used in establishing service-level expectations. The average or median are insufficient to define these expectations, because they don’t deal with probabilities of progressively rarer events that the delivery will take longer. The 80th (one in five will take longer), 85th, 90th, 95th, 98th and 99th percentiles are often used for this purpose. These percentiles can be taken from lead time data sets or calculated from Weibull distribution “navigation tables” as multiples of the average (the N-th percentile-to-average ratio depends only on the shape parameter).

The upper control limit. For the purposes of statistical process control and identifying outliers (special-cause variation), we can establish an upper control limit. In creative industries, it is not necessary to establish control limits by calculation. Collaborative inquiry — this project took X days to deliver, do we agree on a single obvious cause that led to the delay? — can be used instead to differentiate between special- and common-cause variation.

If we need to calculate the upper control limit, then adding three standard deviations to the average is plainly incorrect, because the lead time distribution is never Gaussian. With Weibull distribution, we can set the limit at the same probability as being within the average plus three sigma on the Gaussian, which is 99.865%.  Credit to Bruno Chassagne who took this approach assuming the Exponential distribution in IT Operations work and presented the results at Lean Kanban Benelux 2011 conference.  Owing to the properties of Weibull distribution, the control limit is proportional to the lead time average and depends only on the shape parameter. In product development, the ratio of 4 to 6 is common, in operations and customer care, 10 to 12.


Different points on a lead time distribution chart play different roles. Understanding those roles can help us:

  1. assess service delivery capabilities
  2. set service-level expectations
  3. create delivery forecasts
  4. create short feedback loops
  5. understand workers’ biases
  6. manage variation
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Analyzing the Lead Time Distribution Chart

This post is about one of the key measurements of flow we use in Kanban: lead time. We’ll talk a bit about how to measure it and analyze and use the results.

Example: a work item on a Kanban board with the start time marked

Example: a work item on a Kanban board with the start time marked

Loosely defined, the lead time is how long it takes a work item to get through the system. There are several variations of this definition. The customer lead time is essentially from concept to cash. In non-manufacturing, knowledge-work kanban systems, we also use the kanban system lead time, measured from the kanban system’s first commitment point to the first infinite buffer.

We also use the term cycle time. In knowledge-work kanban systems, it is always local and qualified where it is to and from. There is no “the cycle time.” There are interesting reasons behind these definitions, but they are beyond the scope of this post.

Example: a policy card on a Kanban board with a reminder to mark lead time

Example: a policy card on a Kanban board with a reminder to mark lead time

Measuring lead time is very easy. All it takes is to note when the work item started, when it was delivered, and subtract the difference. We can do this using timestamps from electronic work management systems. If we use physical Kanban boards, we can also do this by marking the start and end dates in the bottom corners of the card. The “Robinson Crusoe method”, placing a mark on the card every day, proved ineffective in the modern office.

After delivering several items, we’ve got a lead time data set. We can now plot a histogram and analyze the distribution


Here is an example of lead time distributions from an organization developing custom IT solutions.

Histogram showing the distribution of lead times of projects

This is not an uncommon lead time distribution shape, with a left-shifted asymmetric “hump” and a long, fat “tail.” The best fit for this data set is the so-called Weibull distribution with the shape parameter 1.62. (Weibull is a parametrized family of distributions. Varying the shape parameter tweaks the distribution curve into several distinct shapes. But this is a topic for another post.)

The delivery takes 34 days on average. Eighty-five percent of solutions are delivered in 61 days, 98% take up to 96 days. We know these probabilities if we know nothing else about a project.

Drilling Down

This organization’s projects are not all the same. They deliver solutions of several different types. The following histogram shows the lead time distribution of four major project types. (For reasons of anonymity, these project types are only identified by colour-coding.)

A lead-time histogram colouring four major types of projects

We can see on this chart that each project category has a different distribution. Blue implementations typically take longer to deliver than green projects. There are several, disproportionally too many red reports in the distribution’s tail, even though their distribution peak is left-shifted. Purple integrations take quite a long time even in the best case.

The first bits of information known about a project is which of the categories it falls into. In Kanban, we call these work item types. This information is clear, unambiguous, and requires no time or effort to get. Let’s drill down our lead time data set by work item type.

Here is the lead time histogram for “green” projects which account for a bit more than half of all deliveries.

The lead time histogram, including only one type of projects

Green projects are delivered slightly faster than the average for all projects, in 30 days vs. 34. The 85th percentile of deliveries also take less time, 54 days vs. 61; and the 98th percentile takes 85 days vs. 96.

The best distribution fit for green projects is, again, Weibull with shape 1.62. One important quality of this distribution is that the percentiles (the 85th, the 98th and others) depend only on the average and the shape parameter. Anyone can calculate the average. Finding the shape parameter takes some statistical skill, but the good news is, it doesn’t change much with time. (I plan to write later about simpler ways to find the shape parameter with sufficient precision.) Knowing the shape parameter and the average, we can calculate the 85th, the 98th and any other percentile we need. We can also, of course, take these percentiles from the data set.

Let’s now take a look at red reports.

Lead time distribution histogram for another work item type, showing a different distribution pattern

The delivery process company uses for this type of project is different and this shows in the shape parameter of the best-fit distribution. It is again Weibull, but the shape parameter is much smaller: 1.23. The average delivery takes 35 days, the mode (most common lead time ) is much less, 85% of deliveries take 61 days, 98% take 96 days.

Looking at other work item types, blue implementation projects follow a distribution similar to green projects: shape parameter 1.65, average 40 days, 85% delivered in 66 days, 98% in 81 days.

The organization’s process for delivering purple integration projects is the most regimented and uses several phase gates. This is also reflected in the lead time distribution shape (shape parameter: 3.22). It is more symmetrical, with very few “small” (short lead-time) projects. The average delivery time is 56 days, 85% in 70 days, 98% in 100 days.

Establishing Service-Level Expectations

We can now establish service level expectations for different work item types delivered by this organization. The following table summarizes data for four work item types. For the sake of comparison, I included percentile estimates obtained using parametrized distributions. (A little secret: I have “navigation tables” with pre-calculated ratios for all common distribution shapes, so I simply took the average lead time and multiplied it by numbers from the table using my phone’s calculator).

Work item type Shape Average lead time From the lead time data set From the parametrized distribution
85% 98% 85% 98%
Green Projects 1.62 30 54 85 51 83
Red Reports 1.23 35 61 96 63 112
Blue Implementations 1.65 40 66 81 68 110
Purple Integrations 3.22 56 70 100 78 99

Notice that the 98th-percentile estimates obtained using parametric distributions are fairly conservative. This is appropriate as small data sets (there were only 19 Red and 14 Blue work items in the data set) don’t capture probabilities of rare events well.


Lead time is easy to measure. There are several definitions and it’s important to understand where to measure from and to. Many services deliver a mix of work item types. Therefore it is important to drill down our lead time data by work item type, so that separate service level expectations can be established for each. Properties of common lead-time distributions are well-studied and can be used to support the SLEs and forecasting.

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