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> Modified Rule of Six, ...and "always having a shot"
GunnerJ
post Apr 8 2005, 12:10 AM
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QUOTE
It's true, but providing that extra successes still mean a better overall quality of success, it provides a chance for a better lucky shot, as far as I can tell.


True. True. I just think that we can fairly modify the word "better" here with either "negligably" or "insignifigantly in most cases" when it comes to the nRo6.
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Ellery
post Apr 8 2005, 01:18 AM
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It is better to do this analytically, but it's not quite trivial.

The probability of getting at least N successes on one roll is

> q(N) (1/3)*(1/6)^(N-1)

because you have to roll a 6 (N-1) times and then you can roll a 5 or 6 to finish off. Therefore, the chance of getting exactly N successes is

> p(N) = q(N)-q(N+1) = (1/3)*(1/6)^(N-1) - (1/3)*(1/6)^N = (1/3)*(5/6)*(1/6)^(N-1).

Therefore the expected number of successes E you will get is sum( N*p(N) , N=1..infinity), which we can rewrite as

> E = (1/3)*(5/6)*( 1 + 2*(1/6) + 3*(1/6)^2 + ... + N*(1/6)^(N-1) + ... ).

This is a standard infinite series of the for n*x^(n-1). If we note that dx^n/dx = n*x^(n-1), we can rewrite as (1/3)*(5/6)*( sum( d(x^n)/dx ) )|x=(1/6). Pull out the derivative to get

> E = (1/3)*(5/6)* (d/dx)[ sum(x^n,n=1..inf) ] | x=(1/6)

we then note that sum(x^n,n=1..inf) = 1/(1-x) - 1, and that (d/dx)[ 1/(1-x) ] is 1/(1-x)^2, and we have

> E = (1/3)*(5/6)/(5/6)^2 = (1/3)/(5/6) = 2/5 = 40%.

Since each die is independent, the expected number of successes for n dice is

> E(n) = 2*n/5

In contrast, for the non-exploding case, the expected number F(n) is

> F(n) = n / 3

You can also compute the probability distributions of each method to compute the probability of getting a certain number of successes with or without exploding dice. The scope of that computation is beyond what I want to explain here right now, but I'm happy to produce tables upon request. Just as an example, if you're rolling four dice, the probabilities are (without and with exploding dice)

> 0 successes: 19.75% 19.75%
> 1 success: 39.51% 32.92%
> 2 successes: 29.63% 26.06%
> 3 successes: 9.86% 13.49%
> 4 successes: 1.23% 5.32%
> 5 successes: 0% 1.76%
> 6 successes: 0% 0.51%
> 7 successes: 0% 0.14%
> 8+ successes: 0% 0.05%
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GunnerJ
post Apr 8 2005, 01:32 AM
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QUOTE
The scope of that computation is beyond what I want to explain here right now, but I'm happy to produce tables upon request.


That would be pretty rocking, actually.
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Ellery
post Apr 8 2005, 02:32 AM
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Aha, there are code tags. That will make this much prettier. How big do you want the tables? I'm doing 12x8 for now, just to keep things in a reasonable amount of space, and truncating to three decimal places. All values are in percentages, of course.

Chances of getting at least N successes, no exploding dice:
CODE

     #0#    #1#    #2#    #3#    #4#    #5#    #6#    #7#    #8#  (succ)
+1+  100    33.333  0      0      0      0      0      0      0
+2+  100    55.556 11.111  0      0      0      0      0      0
+3+  100    70.37  25.926  3.704  0      0      0      0      0
+4+  100    80.247 40.741 11.111  1.235  0      0      0      0
+5+  100    86.831 53.909 20.988  4.527  0.412  0      0      0
+6+  100    91.221 64.883 31.962 10.014  1.783  0.137  0      0
+7+  100    94.147 73.663 42.936 17.33   4.527  0.686  0.046  0
+8+  100    96.098 80.491 53.178 25.865  8.794  1.966  0.259  0.015
+9+  100    97.399 85.693 62.282 34.969 14.485  4.242  0.828  0.097
+10+ 100    98.266 89.595 70.086 44.074 21.313  7.656  1.966  0.34
+11+ 100    98.844 92.485 76.589 52.744 28.9   12.209  3.863  0.882
+12+ 100    99.229 94.605 81.888 60.693 36.848 17.772  6.645  1.876
(dice)


Chance of getting at least N successes, exploding dice:
CODE

     #0#    #1#    #2#    #3#    #4#    #5#    #6#    #7#    #8#  (succ)
+1+  100    33.333  5.556  0.926  0.154  0.026  0.004  0.001  0
+2+  100    55.556 18.519  4.63   1.029  0.214  0.043  0.008  0.002
+3+  100    70.37  33.333 11.728  3.412  0.883  0.211  0.048  0.01
+4+  100    80.247 47.325 21.262  7.773  2.453  0.697  0.183  0.046
+5+  100    86.831 59.396 31.962 14.053  5.273  1.752  0.53   0.149
+6+  100    91.221 69.273 42.753 21.821  9.494  3.629  1.251  0.396
+7+  100    94.147 77.077 52.893 30.488 15.048  6.51   2.524  0.893
+8+  100    96.098 83.092 61.957 39.467 21.691 10.463  4.511  1.768
+9+  100    97.399 87.644 69.761 48.264 29.083 15.438  7.324  3.151
+10+ 100    98.266 91.04  76.288 56.518 36.852 21.281 11.003  5.155
+11+ 100    98.844 93.545 81.623 63.997 44.654 27.772 15.51   7.856
+12+ 100    99.229 95.376 85.902 70.585 52.2   34.657 20.745 11.28
(dice)


Chance of getting exactly N successes, no exploding dice:
CODE

     #0#    #1#    #2#    #3#    #4#    #5#    #6#    #7#    #8+#  (succ)
+1+  66.667 33.333  0      0      0      0      0      0      0
+2+  44.444 44.444 11.111  0      0      0      0      0      0
+3+  29.63  44.444 22.222  3.704  0      0      0      0      0
+4+  19.753 39.506 29.63   9.877  1.235  0      0      0      0
+5+  13.169 32.922 32.922 16.461  4.115  0.412  0      0      0
+6+   8.779 26.337 32.922 21.948  8.23   1.646  0.137  0      0
+7+   5.853 20.485 30.727 25.606 12.803  3.841  0.64   0.046  0
+8+   3.902 15.607 27.313 27.313 17.071  6.828  1.707  0.244  0.015
+9+   2.601 11.706 23.411 27.313 20.485 10.242  3.414  0.732  0.097
+10+  1.734  8.671 19.509 26.012 22.761 13.656  5.69   1.626  0.34
+11+  1.156  6.359 15.896 23.845 23.845 16.691  8.346  2.981  0.882
+12+  0.771  4.624 12.717 21.195 23.845 19.076 11.127  4.769  1.876
(dice)


Chance of getting exactly N successes, exploding dice:
CODE

     #0#    #1#    #2#    #3#    #4#    #5#    #6#    #7#    #8+#  (succ)
+1+  66.667 27.778  4.63   0.772  0.129  0.021  0.004  0.001  0
+2+  44.444 37.037 13.889  3.601  0.814  0.171  0.035  0.007  0.002
+3+  29.63  37.037 21.605  8.316  2.529  0.672  0.164  0.038  0.01
+4+  19.753 32.922 26.063 13.489  5.32   1.756  0.514  0.138  0.046
+5+  13.169 27.435 27.435 17.909  8.78   3.521  1.222  0.381  0.149
+6+   8.779 21.948 26.52  20.932 12.327  5.865  2.379  0.854  0.396
+7+   5.853 17.071 24.183 22.405 15.441  8.538  3.986  1.63   0.893
+8+   3.902 13.006 21.135 22.49  17.776 11.228  5.952  2.743  1.768
+9+   2.601  9.755 17.883 21.496 19.182 13.645  8.113  4.173  3.151
+10+  1.734  7.226 14.752 19.77  19.666 15.571 10.278  5.847  5.155
+11+  1.156  5.299 11.922 17.626 19.343 16.882 12.261  7.655  7.856
+12+  0.771  3.854  9.474 15.317 18.386 17.543 13.912  9.465 11.28
(dice)


From the previous analysis, we know that with exploding dice we expect 0.4 successes per die, and without we expect 0.3333.... This means that the "12" line for non-exploding dice should average 4 successes, and the "10" line for exploding dice should also average 4 successes. Keep this in mind when comparing rolls--it's fairer to compare sets of dice with the same average.
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Phantom Runner
post Apr 8 2005, 04:14 AM
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AH!! For the love of Nuyen!!
Let's get back to the topic of the post....

QUOTE

And yeah, as more and more people are coming up with this themselves, I'm starting to think this would be a very good idea --


Actually this idea is not new at all. Remember it being stated that the new SR mechanic is based on the nWoD dice rolling mechanic except that SR will use d6s?
Well in nWoD they have the 10 again rule, which states 10s count for their success and then the player gets to roll them again. Each 10 that comes up counts again and gets a reroll again....

For SR, "6 again" might not be a bad idea. But I don't know (as few do) if the devs will want to make the SR mechanic so similar to the nWoD mechanic (not that that's a bad thing). Rather I see a player being able to get dice from a convention SR3 already has called a "Karma Pool". Often times a player only had 1 or 2 dice and really needed to get more than 1 or 2 successes. By spending Karma they could increase their dice pool....


...At least I hope Karma Pool is still in the game and works the same way...
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Ellery
post Apr 8 2005, 04:42 AM
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The above stuff shows how much of a shot someone has under different conditions. How is this not exactly the topic of the post?

Anyway, no, it's not really new, but then again, it's not new in nWoD because SR was doing it before (albeit structured slightly differently). I'd bet there was some game doing it before SR was doing it. Heck, there are even entries like "20 - roll again twice" in 1st ed. AD&D treasure tables. Maybe we can find some classic dice games where you get rerolls to rack up success....

Regardless of how old the idea is and who thought of it first, it still makes sense here, I think.
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Critias
post Apr 8 2005, 05:54 AM
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Dammit. Who got her started? Who was it?! You've doomed us all! The math-speak, it consumes my soul !!
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mfb
post Apr 8 2005, 06:03 AM
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she blinded me with advance probability maths.
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otaku mike
post Apr 8 2005, 02:03 PM
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Having a "10 again" rule in nWoD is very different than a "6 again" rule in SR, for a reason none of you have mentioned yet: time.
You will reroll a D10 less often than a D6. The developers of SR4 are trying to streamline the game and make it faster/more fluid. Rerolling dice takes time. You roll, then figure out which ones are the 6, then roll again, and probably roll again once. That considerably add time to the whole process. For that reason only, I think they might avoid using that rule. I could be wrong though.
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GunnerJ
post Apr 8 2005, 02:14 PM
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It's hard to make sense of the statistics without knowing how the successes will be used. However, given what I know of both Shadowrun and Exalted, I will make the guess that there are (at least) two possible uses for successes: a threshold system in which you must get X or more successes to pass a test, or a more open ended system where the number of successes you get determines how well you do.

Given these assumptions, let's look at three results: One success, which is either an easy task in a threshold test or just scraping by in the open-ended system; three successes, which is either a moderately difficult task as a threshold or a pretty good result on an open-ended success test; and five successes, which might be a very hard or near-impossible threshold or a superlative performance on an open-ended test.

The goal of the nRo6 is to make it theoretically possible to succeed at any threshold/level with any number of dice while still not breaking the system of difficulty by granting a glut of extra successes. I think it's safe to say, looking at Ellery's charts, that nRo6 works as it should for the three conditions I outlined. Let's look at the average roll (6D6) and the elite roll (12D6).

Without exploding dice, an average roller can get an easy task/just passing result (one or more successes) in 9/10 rolls, and an elite roller only fails once out of hundred rolls. This is reasonable. For a moderately challenging task/good performance (three or more successes), an average roller can get it a little under a third of the time, while a elite roller gets it 4/5 times. This is a little steep for the average roller, but we are talking about good results or a fairly high bar, which one would expect only someone very well trained and/or talented to get with any regularity. For a near impossible task/outstanding performance (five or more successes), the average roller can get it maybe once every fifty rolls, while the elite roller gets it a third of the time. Five is supposed to be very hard to achive, so someone who's just average should only get that many successes rarely, while for someone of extraordinary ability, getting it should be possible but not routinely expected, and this is what we see.

Now, let's look at rolls with the nRo6. An easy/basic result is achived with the same regularity by both average and elite rollers as without exploding dice. For a challenging/good result, average rollers succeed 2/5ths of the time, while elite rollers fail only three times out of every twenty rolls (i.e., 17/20 times they succeed). This is only modestly more than without rerolls, and presrves the "infrequent for grunts, common for experts" balance. For a near impossible/extraordinary result, average rollers get it 1/10th of the time and elite rollers get it half the time.

That's where we see some difficulty. The chance of an average roller getting an outlandishly hard task has been quintupled, and elite rollers see a 20% increase. This follows the theme from my results on the first page that show that the average successes increases noticably with very high dice. This may or may not be a bad thing, however: 10% is still fairly rare, and when the situation is critical, elites would probably rather not be flipping a coin to see if they succeed. Still, if it is a problem, the very hard/excellent performance bar can be moved to six or more successes, which has results very close to those for five or more successes without exploding dice.
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GunnerJ
post Apr 8 2005, 02:16 PM
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QUOTE (otaku mike @ Apr 8 2005, 02:03 PM)
Having a "10 again" rule in nWoD is very different than a "6 again" rule in SR, for a reason none of you have mentioned yet: time.
You will reroll a D10 less often than a D6. The developers of SR4 are trying to streamline the game and make it faster/more fluid. Rerolling dice takes time. You roll, then figure out which ones are the 6, then roll again, and probably roll again once. That considerably add time to the whole process. For that reason only, I think they might avoid using that rule. I could be wrong though.


Getting six on a D6 is more common than getting ten on a D10, but I've thrown around 12D6 at a TN of 6 and gotten nothing more times than would seem possible in a just universe. So I can tell you, it's not at all common in any objective sense.
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Kagetenshi
post Apr 8 2005, 02:34 PM
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It happens one out of every six times on average on an unweighted die, so it's pretty darn common.

~J
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GunnerJ
post Apr 8 2005, 02:41 PM
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QUOTE (Kagetenshi)
It happens one out of every six times on average on an unweighted die, so it's pretty darn common.

~J

One out of six is signifigantly more common than one out of ten? I don't know about that. I'm also pretty sure that in practice, the frequency of getting sixes is inversely proportional to how important it is that you get sixes; at least, that's been my experience. I understand that this is an appeal to chance and anecdote and not a mathematical truth, but I also saw a lot of people refering to their poor "dice luck" on one thread, and they didn't seem to be kidding. I knew their pain.

In any case, it's never slowed down any of my games to look for sixes and reroll them. The developers have already streamlined the game to an incredible degree by having a fixed TN. They have a good amount wiggle room for potentially slowing it a tiny bit more.
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Ellery
post Apr 8 2005, 02:49 PM
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Er, I think the 6-dice person having as much as a 10% chance to get 5 successes is a feature of the exploding dice system. Also, let's compare apples and apples--a 6d6 normal roller will have as many successes on average as a 5d6 exploding roller. So there it's a little under 2% vs. a little over 5%--a little kinder for achieving that extraordinary result.
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Kagetenshi
post Apr 8 2005, 02:57 PM
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QUOTE (GunnerJ)
One out of six is signifigantly more common than one out of ten?

It is. Over half again as common, in fact.

~J
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GunnerJ
post Apr 8 2005, 03:04 PM
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1/6 = 16.666...%
1/10 = 10%

Again, I'm not so sure.

Look, all I'm saying is, if otaku mike is concerned that having to reroll those incredibly common sixes again, and then again, and then quite possibly again is going to make SR4 a slovenly, cartankerous junker of a game crawling at a molasses pace compared to the sleak, sexy, sonic assassin that is the D10 nWoD system, then I can safely say that he has no reason to fear. In my three years playing Shadowrun, rerolling twice happens every so often in a game, it's a rare game where a die is rerolled three times, and I can count on one hand how many times I've seen a die rerolled more than four times. It was usually an event of some awe and celebration, and I feel that if nothing else, if SR4 doesn't have rerolls, it will be a poorer gaming experience for not having those events.
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GunnerJ
post Apr 8 2005, 03:07 PM
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QUOTE (Ellery)
Er, I think the 6-dice person having as much as a 10% chance to get 5 successes is a feature of the exploding dice system. Also, let's compare apples and apples--a 6d6 normal roller will have as many successes on average as a 5d6 exploding roller. So there it's a little under 2% vs. a little over 5%--a little kinder for achieving that extraordinary result.

I agree, actually. I merely put forward the option of increasing the threshold to six to show how easy it would be to correct if it is indeed necessary to correct it.
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Eyeless Blond
post Apr 8 2005, 03:16 PM
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QUOTE (GunnerJ)
In my three years playing Shadowrun, rerolling twice happens every so often in a game, it's a rare game where a die is rerolled three times, and I can count on one hand how many times I've seen a die rerolled more than four times. It was usually an event of some awe and celebration, and I feel that if nothing else, if SR4 doesn't have rerolls, it will be a poorer gaming experience for not having those events.

Indeed, this is one thing I love about exploding dice; on occasion it makes the die rolls themselves memorable, which is something you rarely see in non-exploding systems. It's just fun seeing everyone's eyes pop out (the roller's included!) when you hit something like a TN 25, and talk about it even days afterward ("Dude, remember when you rolled that 34? That was so weird!") Maybe it *is* a little slower, but IMO it's time well spent, and it's rare enough to not bog down the game.
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Kagetenshi
post Apr 8 2005, 03:21 PM
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Oh, I agree that it's not going to slow the game down. I really don't know what people are talking about when they complain about rerolling sixes or rolling a lot of dice, that always seemed to be the best part to me.

~J
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esmdev
post Apr 8 2005, 03:36 PM
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Wasn't the exploding 6 system was first implemented in SR1?
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Kagetenshi
post Apr 8 2005, 03:40 PM
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Yes, why?

~J
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esmdev
post Apr 8 2005, 11:30 PM
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QUOTE (Kagetenshi)
Yes, why?

Just commenting that it would be more retro Shadowrun than pirated WOD as some people semi-aluded to above. :)
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Kagetenshi
post Apr 8 2005, 11:33 PM
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No one's complaining about the exploding dice, look at SR3. It's the fixed-TN, exploding-dice-for-extra-successes that start looking like WoD.

~J
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Cougaar
post Apr 11 2005, 02:57 PM
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QUOTE (Eyeless Blond)
Indeed, this is one thing I love about exploding dice; on occasion it makes the die rolls themselves memorable, which is something you rarely see in non-exploding systems. It's just fun seeing everyone's eyes pop out (the roller's included!) when you hit something like a TN 25, and talk about it even days afterward ("Dude, remember when you rolled that 34? That was so weird!") Maybe it *is* a little slower, but IMO it's time well spent, and it's rare enough to not bog down the game.

Maybe I'm a bit masochistic when it comes to rolling dice - at the moment I like high target numbers and the "rule of 6" - especially in combat! :D

(from one of the last gaming sessions..)
GM: "Cougi, you're next, but from where you're standing you can't get a safe shot at an enemy."
Me: "So I'll make an unsafe shot. Gimme my TN."
GM: "But two of your chummers are in the way."
Me: "Yepp, double figure TN - gimme!"
GM: "If you miss you'll probably hit your chummers."
Me: "I know.. so what's my TN?"
GM: (sighs) "It's 22."
Me: "Ah.. now that's what I call a target!" (starts rolling dice)
Chums (nerveously stare at dice)
GM (stares at dice)
Me: "Bang.. a success!" 8)
(IP the already injured opponent goes down to my shot)
Chums (start breathing again)
GM: "Okay, you got lucky that time.. who's next?"
Me: "Waitaminute.. it's still my turn! Who else can I shoot?" (grins like a maniac)
Chums (stare at me then at the GM)
GM: "You can't be serious!"
Me: "I sure am.. so next opponent and TN?"
...

I don't try to pull stunts like that all the time.. but now and again, I like the idea of beating seemingly unbeatable odds. It's what makes us PCs special - not that we can do it, but that we try.

Here's to hoping there is something similar in SR4! :D
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RSS Lo-Fi Version Time is now: 25th August 2019 - 12:36 PM

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