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jt512
Jul 15, 2011, 2:15 AM
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I have analyzed all the data in Table 2 in the Pavier paper (from which ptlong2's figure was apparently derived) for which the belay connection was tied off (N=27), in order to investigate whether there is a relationship between fall length and maximum impact force after the the effect of the fall factor is accounted for. There is such a relationship (at least in this dataset), and surprisingly, it's non-linear.
Method: I used multiple linear regression models with maximum impact force as the dependent variable. In the first model, I included as independent variables fall factor, fall length, mass, and carabiner radius (which was coded as a 3-level factor to allow for a possible non-linear relationship with impact force). Regression diagnostics strongly suggested a non-linear effect of fall length on impact force. To account for this I ran a second model in which I added a term for an interaction between fall length and fall factor. Carabiner radius was not significantly associated with impact force, and excluding if from the model had essentially no effect on the results, and so, for simplicity, I excluded it from the final model.
Results: The initial model assumed that if there was an independent relationship between impact force and fall length after controlling for fall factor, mass, and carabiner radius, that the relationship would be linear. Although this model seemed to fit the data well (adjusted R-square=0.984), regression diagnostics indicated that there was a non-linear relationship between fall length and fall factor, and so a linear model was not appropriate. The following figure shows these regression diagnostics (click to enlarge).
The panels show excellent linear relationships with impact force for fall factor and mass, but not for fall length (bottom right quadrant). The linear model (red dashed line) overestimates the impact force when the fall length is low or high, and underestimates the impact force when the fall length is intermediate. This implies that a model in which fall length had a quadratic relationship with impact force independent of the fall factor would be a better fit to the data. But this seems nonsensical. It would mean that, for a given fall factor, both short and long falls would have lower impact forces than middle-sized falls. A more plausible explanation is that the effect of fall length on impact force depends on the fall factor itself, and thus the model should include a term to account for an interaction between fall length and fall factor. The regression diagnostics for this interaction model are shown below (click to enlarge).
Comparing these diagnostic plots with those for the strictly linear model shows the fit is improved for all variables. The fits for mass and fall factor, which were quite good in the linear model, are now essentially perfect, and the fits for fall length (lower left) and the interaction between fall length and fall factor (lower right) are excellent. This model also improved the adjusted R-square to 0.997.
The final fitted regression model is:
impact force = –1.29 + 0.0523*mass + 4.23*ff + 0.353*length – 0.335*length*ff
where mass is in kg, "length" is the length of the fall in meters, and ff is the fall factor. The last term in the model represents the interaction between length and fall factor. All regression coefficients were highly significant (P < 0.00001).
The following scatter plot of the observed and model-predicted impact force vs. fall factor shows how well the model fits the data (click to enlarge). Notice how close the predicted values (red circles) are to the observed values (black discs).
To better understand the interaction between fall length and fall factor, that is, how the effect of fall length on impact force depends on the fall factor, we can factor out the fall length from the last two terms in the previous equation. Doing so allows us to express the regression equation as
impact force = –1.29 + 0.0523*mass + 4.23*ff + (0.353 – 0.335*ff)*length .
This shows that the effect of fall length on impact force is a linear function of the fall factor: for each meter of fall length, the impact force is increased by 0.353 kN less 0.335 kN times the fall factor. Since 0.353 is close to 0.335, the effect of fall length on impact force reverses at a fall factor of about 1. For fall factors less than 1, the longer the fall length, the higher the impact force; for fall factors greater than 1, the longer the fall length, the lower the impact force.
Limitations: In a small dataset such as this, model overfitting is a risk. The results should be considered food for thought rather than definitive.
Conclusion: In this dataset, fall length had a significant non-linear relationship with maximum impact force after the effects of fall factor and falling mass were taken into account. This non-linearity can be modeled as an interaction between fall length and fall factor in which, for fall factors less than 1, the longer the fall is, the higher the impact force is; and for fall factors greater than 1, the longer the fall is, the lower the impact force is. These results should be interpreted with caution due to the limited number of data points available for analysis.
Jay
(This post was edited by jt512 on Jul 15, 2011, 3:58 AM)
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dagibbs
Jul 15, 2011, 2:37 AM
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Thanks for the very thorough analysis, Jay.
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shockabuku
Jul 15, 2011, 2:12 PM
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Thanks for the analysis. Unfortunately I can't see your plots right now - I think your server is down.
I wonder if the results have to do with friction at the biner (top piece) not allowing the belay side of the rope to interact as well as the climber's side.
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cracklover
Jul 15, 2011, 3:29 PM
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Jay, you lost me at one of your early graphs. Could you explain what you're plotting in this one labeled Component+Residual(force) by Length?
Thanks!
GO
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jt512
Jul 15, 2011, 4:07 PM
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cracklover wrote: Jay, you lost me at one of your early graphs. Could you explain what you're plotting in this one labeled Component+Residual(force) by Length? Thanks! GO
Each of the "Residual + Component" plots shows the relationship between impact force and the variable on the horizontal axis, after controlling for the effects of all other independent variables in the model. So, what the panel for Length in the first set of plots is showing is that after controlling for the effects of the other terms in the model, the relationship between impact force and fall length is curvilinear.
Jay
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ptlong2
Jul 15, 2011, 5:33 PM
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Nice work, Jay.
I had originally noticed that there seemed to be a trend in the fall length to tension numbers but suspected that it was too small to be above the noise. I'm still not convinced this isn't the case. You may simply have found a nice mathematical fit that has no greater significance. It's hard to say without more data.
What physical interpretation is there for a reduced tension with increasing fall length for FF > 1?
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jt512
Jul 15, 2011, 6:00 PM
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ptlong2 wrote: Nice work, Jay. I had originally noticed that there seemed to be a trend in the fall length to tension numbers but suspected that it was too small to be above the noise. I'm still not convinced this isn't the case. You may simply have found a nice mathematical fit that has no greater significance. It's hard to say without more data. What physical interpretation is there for a reduced tension with increasing fall length for FF > 1?
Beats me. The effect is small, but it's there. The only reason we can see is that there is so little noise in the data. In fact, the data seem almost too good to be true. The effect may have something to do with the design of the experiment, or who knows, it may be real and generalizable to experiments involving solid falling masses with fixed belay.
Jay
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jt512
Jul 15, 2011, 9:43 PM
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ptlong2 wrote: jt512 wrote: The effect is small, but it's there. The only reason we can see is that there is so little noise in the data. In fact, the data seem almost too good to be true. The effect may have something to do with the design of the experiment, or who knows, it may be real and generalizable to experiments involving solid falling masses with fixed belay. I'll bet it's an artifact of some sort. Here are some more data, taken from a paper that is mostly about static ropes but includes drops with a 10.6mm dynamic. "Fall Factors & Life Safety Ropes: a closer look" Chuck Weber, PMI Quality Manager ITRS 2001 
Looking at just the dynamic rope data, there appears to be a non-linear relation between fall length and impact force, as there was in the Pavier data. In the Pavier data, there were two models that explained the data equally well from a statistical perspective: a quadratic relationship between fall length and impact force that was independent of fall factor, or an interaction between fall length and fall factor. I chose the latter because I thought the former was implausible. Why, given the fall factor, would intermediate length falls produce higher impact forces than both shorter and longer falls? But this is what we see in the PMI data. There is a slight drop off in the impact force for the longest fall in the first three panels.
With just fall factor and fall length in the model, the adjusted R-squared for the PMI data is 0.951. Adding a quadratic term for fall length increases the adjusted R-squared to 0.963, with a p-value for the quadratic term of 0.049.
So we now have two datasets that show a non-linear relationship between fall length and impact force that can be modeled as a quadratic effect that is independent of fall factor.
Jay
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ptlong2
Jul 15, 2011, 11:11 PM
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jt512 wrote: Why, given the fall factor, would intermediate length falls produce higher impact forces than both shorter and longer falls? But this is what we see in the PMI data. There is a slight drop off in the impact force for the longest fall in the first three panels.
Maybe, but it could simply be noise. Look at the data for the low stretch ropes. Sometimes one goes up and the other down, then it's reversed for a different case. What is the most likely explanation for that?
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jt512
Jul 15, 2011, 11:38 PM
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ptlong2 wrote: jt512 wrote: Why, given the fall factor, would intermediate length falls produce higher impact forces than both shorter and longer falls? But this is what we see in the PMI data. There is a slight drop off in the impact force for the longest fall in the first three panels. Maybe, but it could simply be noise. Look at the data for the low stretch ropes. Sometimes one goes up and the other down, then it's reversed for a different case. What is the most likely explanation for that?
I don't know. I don't think we should bring low-stretch ropes into the analysis, either formally or informally. I wouldn't assume a priori that the results of low-stretch ropes should be applicable to dynamic ropes.
I suspect that the effect observed in the Pavier data could have something to so with their methodology. As far as I can tell, they performed multiple drops on the same rope until failure, and recorded the highest impact force observed on any of those drops. That's different than performing one drop per rope sample, and recording the maximum impact force. There are likely to be more variables that affect when a rope breaks after multiple drops than affect what the maximum impact force is on a single drop, and therefore I think it's plausible that there could be more a complicated relationship between fall length and impact force in the former case than in the latter.
Jay
(This post was edited by jt512 on Jul 16, 2011, 10:13 PM)
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ptlong2
Jul 16, 2011, 12:34 AM
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jt512 wrote: ptlong2 wrote: Look at the data for the low stretch ropes. Sometimes one goes up and the other down, then it's reversed for a different case. What is the most likely explanation for that? I don't know. I don't think we should bring low-stretch ropes into the analysis, either formally or informally. I wouldn't assume a priori that the results of low-stretch ropes should be applicable to dynamic ropes.
The reason I mentioned it was to suggest that the accuracy of their data is probably not high enough to give too much credence to the slight downturn in in the plots.
In reply to: I suspect that the effect observed in the Pavier data could have something to so with their methodology. As far as I can tell, they performed multiple drops on the same rope until failure, and recorded the highest impact force observed on any of those drops. That's different than performing one drop per rope sample, and recording the maximum impact force. There are likely to be more variables that affect when a rope breaks after multiple drops than affect what the maximum impact force is on a single drop, and therefore I think it's plausible that there could be more a complicated relationship between fall length and impact force in the former case than in the latter.
Who knows what Pavier did? Look at Powick's results: there's one case where the first drop peak is lower for the shorter fall but the maximum peak is lower for the longer fall.
Do you really find the evidence for a drop-off in tension statistically compelling?
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jt512
Jul 16, 2011, 2:44 AM
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ptlong2 wrote: jt512 wrote: ptlong2 wrote: Look at the data for the low stretch ropes. Sometimes one goes up and the other down, then it's reversed for a different case. What is the most likely explanation for that? I don't know. I don't think we should bring low-stretch ropes into the analysis, either formally or informally. I wouldn't assume a priori that the results of low-stretch ropes should be applicable to dynamic ropes. The reason I mentioned it was to suggest that the accuracy of their data is probably not high enough to give too much credence to the slight downturn in in the plots. In reply to: I suspect that the effect observed in the Pavier data could have something to so with their methodology. As far as I can tell, they performed multiple drops on the same rope until failure, and recorded the highest impact force observed on any of those drops. That's different than performing one drop per rope sample, and recording the maximum impact force. There are likely to be more variables that affect when a rope breaks after multiple drops than affect what the maximum impact force is on a single drop, and therefore I think it's plausible that there could be more a complicated relationship between fall length and impact force in the former case than in the latter. Who knows what Pavier did? Look at Powick's results: there's one case where the first drop peak is lower for the shorter fall but the maximum peak is lower for the longer fall. Do you really find the evidence for a drop-off in tension statistically compelling?
I haven't really looked at Powick's results. But, at this point, between the Pavier data and the PMI data (including the static and low-elongation rope data), I'd say that the evidence favors the existence of a non-linear relationship between fall length and impact force after the effect of fall factor is taken into account. Looking at the PMI data, the overall impression is that the effect of fall length decreases as the fall length increases. Whatever is going on in the Pavier data, it isn't random. He has hardly any random error in his data at all. The pattern in the lower right panel in my first set of residual plots is almost certainly a systematic departure from linearity. What caused it, I don't know. It could be systematic error or something meaningful in the context of Pavier's methodology, but not elsewhere.
Edit: The residual effect of fall length on impact force was observed because the model wrongly assumed that there should be a linear relationship between maximum impact force and fall factor. See my next post.
Jay
(This post was edited by jt512 on Jul 16, 2011, 10:15 PM)
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jt512
Jul 16, 2011, 7:38 PM
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ptlong2 wrote: shockabuku wrote: I would still like to see some data on a case where these other factors aren't involved. These data are taken from a paper by Martyn Pavier. The selected drops were performed with a 70kg steel mass and the belay was tied off. It appears that with the "other factors" removed the length of fall has no significant effect on maximum tension, at least over this limited range.  For real life falls I would wonder about the effects of belay device and belayer behavior.
So, this ^^^^ turned out to be a red herring that sent me on a wild goose chase of mixing metaphors and looking for a residual effect of fall length on impact force. Why would you expect there to be a linear relationship between fall factor and maximum impact force in the first place? The standard relationship is given by
where F is the impact force, w is the climber's weight, k is the rope modulus, and r is the fall factor. Pavier's observed impact forces are almost perfectly correlated with the values predicted from this model.
Using linear regression, I compared the impact forces observed by Pavier with those predicted by the standard model above (Pavier, Table 2, Belay condition = "tied off", N=27). It was difficult to tell from Pavier's paper what value should be used for the rope modulus k, and the fit depends slightly on the value chosen. Using k = 30.9 (equivalent to a rope with an impact force rating of 10.1 kN), results in a regression coefficient, or slope, of 1 and an intercept of –0.62; in other words, the standard model overestimates the impact force by a constant 0.62 kN. This model almost perfectly fits the data, as assessed by an r-squared value of 0.995.
Interestingly, cracklover sent me a copy of an analysis he did that did show a residual effect of fall length after accounting for the impact force explained by the standard model, but I'm not seeing it in my analysis.
Jay
(This post was edited by jt512 on Jul 26, 2011, 5:27 AM)
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ptlong2
Jul 18, 2011, 6:46 PM
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jt512 wrote: Why would you expect there to be a linear relationship between fall factor and maximum impact force in the first place?
Because it looked linear! It is curious and humbling how both of us were so easily duped by my blunder. We both knew it was a square root relationship. If only we'd taken a moment to look at Figure 1 in your paper on the subject.
Sorry for the wild goose chase.
It's not all wasted effort though. Your comparison with the standard model might be of interest. We don't have the hang tag specs for the rope which the standard model methodology calls for, but if his data are so good I'm curious why you didn't use Pavier's own 80kg factor 1.77 drop data for the maximum impact force?
Since the friction-adjusted model in your paper predicts even higher tensions it could be that by leaving out some other factor it is "correcting" in the wrong direction.
Maybe this discrepency is accounted for by Pavier's model. I could have sworn I had investigated this and determined that an effective modulus could be used for the ff to peak tension relationship. But I can't find this work so perhaps I'm thinking of something else. It isn't terribly difficult to code Pavier's model, at least not for the simple one-runner 180 degree angle case.
EDIT: I found a piece of what I'd done before, a graph that compared belayer-side tension to fall factor for the following models: standard, Attaway's (aka friction-adjusted), Attaway's corrected for his approximation, and Pavier's. The Pavier model diverged significantly from all of the other three.
(This post was edited by ptlong2 on Jul 18, 2011, 7:23 PM)
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cracklover
Jul 18, 2011, 7:41 PM
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ptlong2 wrote: jt512 wrote: Why would you expect there to be a linear relationship between fall factor and maximum impact force in the first place? Because it looked linear! It is curious and humbling how both of us were so easily duped by my blunder. We both knew it was a square root relationship. If only we'd taken a moment to look at Figure 1 in your paper on the subject. Sorry for the wild goose chase. It's not all wasted effort though. Your comparison with the standard model might be of interest. We don't have the hang tag specs for the rope which the standard model methodology calls for, but if his data are so good I'm curious why you didn't use Pavier's own 80kg factor 1.77 drop data for the maximum impact force? Since the friction-adjusted model in your paper predicts even higher tensions it could be that by leaving out some other factor it is "correcting" in the wrong direction. Maybe this discrepency is accounted for by Pavier's model. I could have sworn I had investigated this and determined that an effective modulus could be used for the ff to peak tension relationship. But I can't find this work so perhaps I'm thinking of something else. It isn't terribly difficult to code Pavier's model, at least not for the simple one-runner 180 degree angle case. EDIT: I found a piece of what I'd done before, a graph that compared belayer-side tension to fall factor for the following models: standard, Attaway's (aka friction-adjusted), Attaway's corrected for his approximation, and Pavier's. The Pavier model diverged significantly from all of the other three.
Well, I was chasing no such geese in my analysis. I was completely comparing apples to apples. Let me know if you'd like me to put this stuff online.
GO
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cracklover
Jul 18, 2011, 8:38 PM
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Edits inserted in red
Here you go, I put it in an online document: https://spreadsheets.google.com/...E&hl=en_US#gid=0
The number for the model_tension comes by plugging all the numbers given in pavier's chart into the standard rope model equation. The only number not provided in his paper is the modulus for the rope. For that I simply assumed a modulus that would provide an average difference of zero. Any other modulus would work equally well, it makes no difference.It turns out that while small changes in modulus make no difference, as the modulus gets over ten, the effect I found disappears. If one assumes a still higher modulus, the affect actually reverses!
So, for pavier's setup, it appears that the shorter the fall, the higher the force, relative to the standard model, and the longer the fall, the lower the force (again, relatively).Again, only at the modulus I used.
Note that this observed difference is quite small! The largest difference is a few percent.
So, to answer the OP's question, at least in Pavier's setup, the FF is almost entirely predictive of the impact force, but there is also a very small (at most a couple percent of the force) change depending on the actual fall length.There is no strong evidence for any relationship between fall distance and force, except as it relates to fall factor.
GO
(This post was edited by cracklover on Jul 19, 2011, 3:29 PM)
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cracklover
Jul 18, 2011, 8:49 PM
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jt512 wrote: The standard model ... almost perfectly fits the data, as assessed by an r-squared value of 0.995.
Interestingly, cracklover sent me a copy of an analysis he did that did show a residual effect of fall length after accounting for the impact force explained by the standard model, but I'm not seeing it in my analysis. Jay
Well you certainly won't see it in that graph, because, as you say, his data almost perfectly fits the model. It's only if you look at the delta force between what the model predicts and his data shows that you can see that at any given FF there is a small but real difference. As I mentioned in my post above, the difference is at most a few percent.
Edited to add the graph inline for those who just want to see the result:
GO
(This post was edited by cracklover on Jul 18, 2011, 9:02 PM)
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dynosore
Jul 18, 2011, 10:57 PM
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shockabuku wrote: Has anyone seen any data on how the maximum impact force changes (or doesn't change) as fall length increases for a constant fall factor?
One thing rarely discussed is the total energy to be absorbed in a fall. Yes, a 10 foot and 40 foot factor one fall will have very similar peak forces, all other things being equal. But what about the duration of those forces? Maybe a cam starts to skid at 5kn in a bad placement. The 5kn force is only exceeded for say, 0.1 sec in a 10 foot F1 fall, but in a 40 foot fall it's exceeded for several times that long. Maybe that's the difference between the cam skipping all the way out and not. Hope that makes sense, I'm in a bit of a hurry.
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jt512
Jul 19, 2011, 3:15 AM
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cracklover wrote: jt512 wrote: The standard model ... almost perfectly fits the data, as assessed by an r-squared value of 0.995.
Interestingly, cracklover sent me a copy of an analysis he did that did show a residual effect of fall length after accounting for the impact force explained by the standard model, but I'm not seeing it in my analysis. Jay Well you certainly won't see it in that graph, because, as you say, his data almost perfectly fits the model. It's only if you look at the delta force between what the model predicts and his data shows that you can see that at any given FF there is a small but real difference.
By "not seeing it in my analysis," I meant that a plot of the residuals (ie, the observed minus the predicted impact force) from the aforementioned model vs fall length shows no relationship. This is plotted in the following figure. The depicted regression line is not significant (p-value = 0.20).
Jay
(This post was edited by jt512 on Jul 19, 2011, 1:56 PM)
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cracklover
Jul 19, 2011, 3:47 AM
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jt512 wrote: ptlong2 wrote: cracklover wrote: [IMG]http://i53.tinypic.com/miz70i.png[/IMG] Cracklover, since the fall length and fall factor are not independent in Pavier's data it is hard to know whether the effect you think you see is due to one or the other or both. In addition, the effect is rather sensitive to the choice of rope modulus. The following figure shows the difference between the impact force predicted by the standard model and the observed impact force vs fall length, using various values for the rope modulus. The rope modulus is represented by U the rope's UIAA impact force rating (click to enlarge).  Jay
Wow, that's fascinating! Thanks for the second set of eyes. I'll have to look that over.
While a modulus that high is beyond what a new dynamic rope could ever have, Pavier has that key phrase "maximum tension" which suggests that he may be reporting the tension after the rope is nearly shot from multiple repeated drops. I don't know, perhaps a modulus could get that high in those conditions.
GO
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