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# Avoidance Diminishing Returns

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### #1 Riot

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Posted 25 November 2008 - 03:11 PM

Conclusion:

Diminishing returns on avoidance are harsh. This is a graph that shows how DR affects Dodge rating, for example, by Gorek:

If your dodge gets too high, go for some parry if you can. Parry rating remains more expensive than Dodge, so don't actively go out seeking to replace all your tank gear with dodge, to tank gear with parry. In the course of doing so, you may even end up losing avoidance.

Defense is another good, balanced way to boost your avoidance, as it gives you Dodge, Parry, and Miss in one stat. It also gives Block as well.

Original Post:

I am not the go-to guy for calculations of any sort at all, but I wanted to share my observations and hopefully share data regarding avoidance as a stat for tanking.

As I'm sure most of us have noticed, when we mouse over our character panes, the window notes diminishing returns, and those DR becomes painfully obvious when equipping a high avoidance trinket, such as the .

This brings to mind a few questions for me, and I'm sure, eventually others as well, and a sort of conclusion:

When does DR start kicking in? When does it start getting really painfully obvious?

It also seems to bring a few premature conclusions to my mind, at least:

Balance Dodge/Parry - yeah, Parry is looking better as a stat if you have a load of Dodge, and I think I suddenly understand the design philosophy on tanking items a lot better.

How overpowered might high levels of Shield Block rating since it's not on any sort of diminishing returns?

It seems to me you'll start hitting DR on Dodge/Parry both at approximately 20% - any better numbers out there?

The difference between the almost right word and the right word is really a large matter--it's the difference between the lightning bug and the lightning. - Mark Twain

### #2 Moogul

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Posted 25 November 2008 - 03:42 PM

How overpowered might high levels of Shield Block rating since it's not on any sort of diminishing returns?

Unfortunately I'm going to reply with questions, not answers, but are we definitely sure of the above? I think with block as it stands for warriors (scaling well with strength, plus getting multiplied by 1.3 from talents and having critical block), that without diminishing returns on block rating, it could be very powerful - if it's itemized a lot, then going for a large block rating set (ie. aiming at an old-style 'crush-immune' set) could work out quite well.

Another related question, is the +miss from defence on diminishing returns? Otherwise defence could end up being a really nice stat, as it increases your avoidance in a nice balanced way.
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### #3 Fola

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Posted 25 November 2008 - 03:55 PM

I'm not sure whether this helps to answer the question or it is something you already looked over, but here is the link to the DR calculations put up on tankspot (later in the thread there is a circuit equivilancy for visualization purposes).

WotLK Diminishing Returns - Avoidance - TankSpot

I see your going the Block rating/value path at the moment using the trinket from Heroic Violet Hold, I would be interested in scoping some of your 25 man and 10 man WWS parses from SC if you have them handy.

edit: My gut feeling is the lack of DR on block will make it a very attractive stat gearing through Northrend.

### #4 Riot

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Posted 25 November 2008 - 04:16 PM

I'm not sure whether this helps to answer the question or it is something you already looked over, but here is the link to the DR calculations put up on tankspot (later in the thread there is a circuit equivilancy for visualization purposes).

WotLK Diminishing Returns - Avoidance - TankSpot

I see your going the Block rating/value path at the moment using the trinket from Heroic Violet Hold, I would be interested in scoping some of your 25 man and 10 man WWS parses from SC if you have them handy.

edit: My gut feeling is the lack of DR on block will make it a very attractive stat gearing through Northrend.

Well, I'm definitely experimenting. You'll note I have decent levels of Expertise/Hit/Stam as well. I try to be well-rounded, and if I need to swap gear to min-max on certain encounters, I'll certainly do that as well.

Thanks, but I was mainly looking for other observations like: "Yeah, about 20%".
The difference between the almost right word and the right word is really a large matter--it's the difference between the lightning bug and the lightning. - Mark Twain

### #5 Jamor

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Posted 25 November 2008 - 05:05 PM

Well, I'm definitely experimenting. You'll note I have decent levels of Expertise/Hit/Stam as well. I try to be well-rounded, and if I need to swap gear to min-max on certain encounters, I'll certainly do that as well.

Thanks, but I was mainly looking for other observations like: "Yeah, about 20%".

Making things like the Esscense of the Gossamer, the Shield Block trink from Voilet Hold Heroic, and the Godly JC trinket much better than they appear. I am working towards a balance in all stats (including tank DPS stats - hit, expt) as well, which has always been my goal in tanking setup.

On a side note, crit blocks with SB up and the SBV trinket popped can be massive. I have seen 6.4K blocks and that was without glyph of blocking.

### #6 Kannojo

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Posted 25 November 2008 - 06:21 PM

This was Phaze's statement on tankspot.

Preliminary numbers show roughly 5% drop in effectiveness per 100 rating.

For example: if 100 dodge rating converts to 5.29% dodge pre-LK, it will convert to ~5.04% dodge in LK, followed by the next 100 rating only giving ~4.78%, and so on.

I'm guessing the Blizzard UI doesn't have the actual formula available, like it does for Armor's mitigation?

This really isn't too much of a value. I'd just suggest you download RatinBuster as there doesn't seem to be an easy metric to track or predict the returns without doing the formula.

### #7 Kannojo

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Posted 25 November 2008 - 07:03 PM

Here are some samples I did in relation to the example I gave my guild when explaining it which can be found here

The amount in parenthesis is the actual dodge from contributing stats (defense/agi/dodge rating) after the diminishing returns are accounted for.The amounts for the dodge is just made up.You would add this to the 5% from anticipation and your base dodge(your character naked). This is a small sample size. If work continues to get boring I'll do the whole thing up to 50% and include parry.

### #8 Gorek

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Posted 25 November 2008 - 07:17 PM

Really quick and really dirty the DR on dodge from GEAR, there is no DR on contributions from talents, base agility, etc. The graphs for parry and miss look similar.

### #9 Kannojo

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Posted 25 November 2008 - 07:53 PM

Anyone find the Cap for Miss ? I cannot find it anywhere. If there's no cap then are there no DR on it ?

### #10 Perakles

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Posted 25 November 2008 - 09:25 PM

Anyone find the Cap for Miss ? I cannot find it anywhere. If there's no cap then are there no DR on it ?

World of Warcraft - English (NA) Forums ->

There's a blue post specifically mentioning DR for miss, so I assume that is capped as well.

My gut feeling is that defense is actually going to wind up being the best value in terms of avoidance for warriors. Figure after defense cap you're getting .6% (dodge/parry/miss) avoidance per ~25 defense rating, and .2% block, which is a non-trivial amount of damage reduction with the new block calculations. Defense is also going to spread out your avoidance over the three stats and reduce the impact of diminishing returns.

Of course, my gut feeling doesn't really mean shit, but unfortunately I don't have the math or excel chops to put together a spreadsheet for it.

### #11 Xerophyte

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Posted 26 November 2008 - 01:13 AM

This is somewhat obvious, but we can convert the tank spot notation for avoidance to the form we usually use to describe Armor DR if anyone prefers thinking in terms of that.

$\frac{1}{A_d} = \frac{1}{c} + \frac{k}{A}$

$A_d = \frac{1}{( \frac{1}{c} + \frac{k}{A} )} = \frac{A}{( k + \frac{1}{c} A )}$

As A approaches infinity Ad approaches its cap c, at rate specified by k. This has the added benefit of being defined for no raw avoidance and since the game doesn't crash whenever a tank is naked it seems likely to be a more accurate representation.

For actual analysis our main interest is probably determining how much of an increase in Ad a single point of rating gives at a specific value of A. An approximation of that is to take the derivative of Ad by A and divide by the amount of rating r required to increase A by 1.

$\Delta A_d = \frac{1}{r (k + \frac{2}{c}A + \frac{1}{c^2 k} A^2 ) }$

This seems more than slightly obtuse, so let's take an example. Albert the warrior has +200 parry rating, and +700 defense rating. We know that each point of defense rating increases defense by 1/4.92 and each point of def adds .04% parry before DR. We know that each point of parry rating increases parry percentage by 1/49.2. Albert thus has an A of approximately (200/49.2 + 700/4.92*0.04) = 9.76 percent raw parry. Furthermore we know that for warriors and parry k = 0.956 and c = 47.0. We can plug all this in above to get the new amount of parry Albert gets per rating, which is specifically and pardoning the wall of tex:

$\Delta P_d = \frac{1}{ 49.2 \times (0.956 + \frac{2 \times 9.76}{47.0} + \frac{9.76^2}{47.0^2 \times 0.956}) } \approx 0.0148 \approx \frac{1}{67.7}$

What did all that arithmetic tell us? In short, that at Albert's current level of parry he needs 67.7 points of parry rating per point of gain in parry percentage, compared to the base of r = 49.2. We can also see that in general the amount of rating rA required for one percentage point of actual avoidance at raw avoidance level A corresponds to the known base rating requirement r as

$r_A = r (k + \frac{2}{c}A + \frac{1}{c^2 k} A^2 )$

Note that even at A = 0, meaning no avoidance from gear, we actually need less than r rating per percentage point of gain. Specifically, we need r * k rating per percentage point at gear avoidance 0. The constant k being smaller than 1 essentially sets the avoidance base at very weakly geared rather than naked.

I reserve the right to have screwed up my derivatives slightly, it's been a while and Mathematica's differentiator seems down. It passes my base sanity checks, at any rate.

### #12 Fruffy

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Posted 26 November 2008 - 04:40 PM

Anytime you are calculating the value of avoidance, you must account for the exponentially increasing return on incremental increases in avoidance. IE, the more avoidance you have, the more valuable an additional point of avoidance becomes.

I did some excel work to calculate the amount of dodge rating needed to attain 1% more dodge for various points between 0% and 75% dodge. I then plugged that number into the equation for total damage reduction gained from 1% additional avoidance (stolen from Quigon's OG prot warrior post), did some finger waving and made a graph plotting the total % dmg reduction from 1 additional point of dodge rating vs current total dodge % as read from your character sheet. I plotted this same graph for various values of parry + miss chance. What I found was a very nearly linear decline in the value of dodge until you approached 100% avoidance (dodge+parry+miss) since the return approaches infinity.

To give myself some "landmarks" to go by when considering how to weight different stats, I calculated how "diminished" each stat was at various points. My numbers are for a human with 5/5 anticipation and 5/5 deflection.

Dodge starts at 1.0 value
at 20% dodge we're at 0.9 value per 1 dodge rating
at 29% dodge we're at 0.8 value per 1 dodge rating
at 38% dodge we're at 0.7 value per 1 dodge rating
at 47% dodge we're at 0.6 value per 1 dodge rating (2558 dodge rating)

Parry starts at 1.0 value
at 11% parry we're at 0.9 value per 1 parry rating
at 16% parry we're at 0.8 Value per 1 parry rating
at 20% parry we're at 0.7 value per 1 parry rating
at 25% parry we're at 0.6 value per 1 parry rating (1623 parry rating)

A lot of this is finger waving, but it really helped me to understand where I stand with diminishing returns. I hope it helps some of you.

I actually came here to look for info on teh value of block rating. Do we know there are no diminishing returns on block rating? Block chance that comes from def rating?

### #13 Bakabon

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Posted 26 November 2008 - 04:45 PM

My gut feeling is that defense is actually going to wind up being the best value in terms of avoidance for warriors. Figure after defense cap you're getting .6% (dodge/parry/miss) avoidance per ~25 defense rating, and .2% block, which is a non-trivial amount of damage reduction with the new block calculations. Defense is also going to spread out your avoidance over the three stats and reduce the impact of diminishing returns.

What I am wondering is if strength actually winds up being the best overall damage reduction stat for warriors. At a hit of 5000, 1 block value is approximately equal to 1 dodge rating (assuming mitigation equals avoidance), not counting diminishing returns. And block value actually has exponential returns.

Given that strength is already the best stat for threat, will we eventually see defense capped warriors gem entirely for strength?

### #14 Xav

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Posted 26 November 2008 - 05:06 PM

It would take a high level of gear to reach "uncrushability" and not use any gems/enchants to get there, to let you put in strength gems. I would also think stamina gems may then be the best choice there, because currently at least you sacrifice a lot of health to achieve it.

On that note, with current gear options, you can reach a set that gets you full coverage from dodge+parry+miss+block (after diminishing returns) with full raid buffs taken in to account, and it's barely enough: about 103%. You get to 1700 block value as well, which actually comes out to an average block of ~2600 when you account for critical block and shield block. However the HP hit at the moment to reach that is something like 7k - which is quite a lot.

The set can definitely have its uses depending on the encounter, and is something I will be collecting to have in case.

As for avoidance diminishing returns, you already experience it harshly as low as 20% of an avoidance stat. At 20% parry for example adding the 1.93% from the trinket only added something like 1.2% actual parry.

The math that should be done at this point would be to compare avoidance levels you can obtain with gear/gems/diminishing returns, average damage taken, and then do the same while gearing for block rating instead, as that's going to be the two different choices for reducing damage taken. (This is all, of course, unrelated to stacking stamina to soak more damage)

Edit: There are no diminished returns on Block Rating or Block Rating from defense - the armory is actually incorrect on this. (Ingame, however, it's right, and anyway on the armory it is just a mouse-over 'typo')

### #15 crohnoes

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Posted 27 November 2008 - 05:12 AM

At what point would 30 parry rating give as much avoidance as 30 dodge rating? Block rating is an iffy stat, you get 2.4~ times as much block chance as you do dodge (pre-DR), but (on a typical boss) you're only reducing say 20% of the damage. Block rating scales with gear, and the benefits diminish with how much block value you don't have. Perhaps I'm just being a moron, but these are just my observations. Excuse my fuzzymath, but you'd have to be saturated with dodge and/or block value to see the returns from block rating triumph over an equal amount of dodge rating.

Here are some arbitrary values somebody more 'mathy' can work with: given a warrior with 60% damage reduction from armor, 1400 block value post glyphs, talents, and raid buffs, etc. how much much dodge chance would this hypothetical warrior need for block rating to beat dodge rating in terms of damage mitigation (or avoidance) against hypothetical boss that hits for 20000 pre-armor?

I'm also pretty sure Blizzard would stomp down hard on anyone that comes close to making a gimmick set full of block value and block rating to get past their avoidance cockblocks.

Edit: I should have read Xav's post before writing this.

### #16 Peppah

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Posted 28 November 2008 - 03:04 PM

The ingame tooltip says 'before diminishing returns'. Is this still the case?

According to ratingbuster and my own calculations(which may very well be wrong) the numbers displayed ingame takes dr into account.

I made a very simpledodge-dr calculater based on the calculations in this thread.
I know the dodge calculated from def is a bit off, but the others seem correct enough.

Did i make some other math-mistake or are the tooltips actually correct?

### #17 Kreen

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Posted 28 November 2008 - 10:27 PM

For actual analysis our main interest is probably determining how much of an increase in Ad a single point of rating gives at a specific value of A. An approximation of that is to take the derivative of Ad by A and divide by the amount of rating r required to increase A by 1.

$\Delta A_d = \frac{1}{r (k + \frac{2}{c}A + \frac{1}{c^2 k} A^2 ) }$

This seems more than slightly obtuse, so let's take an example. Albert the warrior has +200 parry rating, and +700 defense rating. We know that each point of defense rating increases defense by 1/4.92 and each point of def adds .04% parry before DR. We know that each point of parry rating increases parry percentage by 1/49.2. Albert thus has an A of approximately (200/49.2 + 700/4.92*0.04) = 9.76 percent raw parry. Furthermore we know that for warriors and parry k = 0.956 and c = 47.0. We can plug all this in above to get the new amount of parry Albert gets per rating, which is specifically and pardoning the wall of tex:

$\Delta P_d = \frac{1}{ 49.2 \times (0.956 + \frac{2 \times 9.76}{47.0} + \frac{9.76^2}{47.0^2 \times 0.956}) } \approx 0.0148 \approx \frac{1}{67.7}$

Cool stuff here Xero.

For the derivative of the deminished avoidance function I get:

$\frac{kc^2}{(kc+A)^2}$

Which, using the same variables, yields:

$0.01435 \approx \frac{1}{69.688}$

Your function might reduce into mine but I'll use mine for the sake of elegance and trying to take a shot at modelling the choice between parry and dodge at different gear levels.

At what point would 30 parry rating give as much avoidance as 30 dodge rating?

This is something I'd been thinking about a lot recently so I thought I'd take a shot at modelling it. The answer is obviously that there are many points where 30 parry would be better than 30 dodge.

Take this general example with illustrative numbers only:

Bob the Warrior has no parry and 20% dodge. This puts him at the point where 1 parry rating is exactly equal in % avoidance as 1 dodge rating. Now say he has the choice to select 3 of either dodge rating or parry rating. He selects the parry first, since they are equal, but now the next point of parry isn't worth as much as the previous, meaning it's not as good as dodge, so he'll select dodge as the following point.

Now comes the part worth thinking about. Since dodge and parry are diminished at different rates, taking that next point of dodge doesn't necessarily mean that he'll want parry again on the third point.

The point at which parry and dodge provide the same % avoidance is when the derivatives of their diminished avoidance functions are equal to one another:

$\frac{kc_p^2}{(kc_p+P)^2}=\frac{kc_d^2}{(kc_d+D)^2}$

We know that k = 0.956, Cp = 47.003525, Cd = 88.129021.

Subing and simplifying:

$\frac{0.956 \times 47.003525^2}{(0.956 \times 47.003525+P)^2}=\frac{0.956 \times 88.129021^2}{(0.956 \times 88.129021+D)^2}$

$D^2 + 168.50D = 3.51P^2 + 315.93P$

For a Warrior with 20% dodge:

$20^2 + 168.50(20) = 3.51P^2 + 315.93P$

$0 = 3.51P^2 + 315.93P-3770.05$

$P = 10.66, -100.54.$

So, for a Warrior with 20% dodge from gear, 1% parry will provide more absolute avoidance than 1% dodge until the Warrior acheives 10.66% parry.

Now to translate that formula to dodge and parry rating:

1% dodge = 39.34799 dodge rating
1% parry = 49.18499 parry rating

$(39.34799^2)D^2+ (168.50)(39.34799)D = (3.51)(49.18499^2)P^2 + (315.93)(49.18499)P$

$D_r^2+ 4.28D_r = 5.48P_r^2 + 10.04P_r$

where $D_r$ and $P_r$ are dodge and parry ratings.

For a Warrior with 20% dodge from gear you'll have 786.96 rating (or agi equivalent)

$(786.96)^2+ 4.28(786.96) = 5.48P_r^2 + 10.04P_r$

$0 = 5.48P_r^2 + 10.04P_r - 622676.73$

$P = 336.17, -338.00$

336.17 parry rating is 6.83% parry so at 20% dodge from gear, you'll choose parry gems until you reach 6.83% parry from gear.

I'm a bit rusty with this stuff as well so let me know if anyone finds any issues with my math.

### #18 Xerophyte

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Posted 29 November 2008 - 05:14 AM

Cool stuff here Xero.

For the derivative of the deminished avoidance function I get:

$\frac{kc^2}{(kc+A)^2}$

$\frac{kc^2}{(kc+A)^2} = \frac{1}{\frac{(kc+A)^2}{kc^2}} = \frac{1}{\frac{k^2c^2 + 2kcA + A^2}{kc^2}} = \frac{1}{(k + \frac{2}{c}A + \frac{1}{kc^2}A)}$

So, yeh, same expression and I feel dumb for not making the simplification. I no doubt had some equally dumb partial approximations along the way, hence difference in result. Likewise, good stuff on the point of inflection but I think you're not correctly adjusting when transitioning to point of inflection for ratings. When at 0 base dodge and parry the two ratings are definitely not equal, as you're indicating - dodge is clearly the better choice for rating at 0, even though looking at percentages you might as well pick either. Mathwise you're missing the interior bit of the chain rule when substituting in the rating in your derivative, unless I miss my guess.

That should be fairly easy to compensate for. Let's assume current parry from gear $P_0$ and current dodge from gear $D_0$, through some combination of stats that we really don't care much about. How much parry rating $P_r$ do we add before equivalency, given that we need $r_P$ rating per parry percentage and $r_D$ dodge rating? Differentiation via the chain rule gives the equivalency when counting ratings as

$\frac{k c_P^2}{r_P \left(k c_P+\frac{P_r}{r_P} + P_0\right)^2}=\frac{ k c_D^2}{ r_D \left(kc_D+D_0\right)^2}$

$\frac{c_P \sqrt{r_D} }{kc_P + \frac{P_r}{r_P} + P_0}=\frac{c_D \sqrt{r_P} }{kc_D+D_0}$

This leads to the not exactly pretty but hopefully correct point of inflection - derived from Mathematica this time

$P_r = \frac{c_P}{c_D} \sqrt {r_d r_p} D_0 - r_P P_0 + c_p k \sqrt{r_P}(\sqrt{r_D} - \sqrt{r_P})$

This passes the obvious sanity tests: the parry rating point of inflection rises with $D_0$ and $r_D$ - representing a high base dodge or a less efficient rating to avoidance conversion for dodge - and falls with $P_0$ and $r_P$.

This is perhaps somewhat hard to penetrate, so I attempted an enlightening graph. Unfortunately, my OOCalc-fu is weak, so you'll have to live with a somewhat confusing graph instead. Sorry.

[e] Oh fine, pre-sleep explanatory edit. Coloured lines in the graph are particular parry percentages - 3, 6, 9 and so on - and the points of inflection for that parry percentage. If the line representing your parry% from gear is negative at your dodge% from gear then you should stack dodge rating over parry rating. If the other way around, you should stack parry rating over dodge rating.

### #19 Kreen

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Posted 29 November 2008 - 06:25 AM

Thanks, I was totally thinking about the 0 dodge 0 parry point in the car on my way home. Nice work.

### #20 Xerophyte

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Posted 29 November 2008 - 02:19 PM

Having slept on things I realize that a much simpler way to express the same thing I did two posts ago is to just assume my $P_r$ is 0 and solve for $P_0$. This gives us the point of inflection in parry from gear at a particular value of dodge from gear directly and is less awful to calculate.

$\frac{k c_P^2}{r_P \left(k c_P + P_0\right)^2}=\frac{ k c_D^2}{ r_D \left(kc_D+D_0\right)^2}$

$P_0 = \frac{ \sqrt {r_d} c_P}{ \sqrt {r_p} c_D} D_0 + c_p k \left(\frac {\sqrt{r_D}}{\sqrt{r_P}} - 1\right)$

Entering numbers into that yields

$P_0 = 0.47704 D_0 - 4.7440$

Slightly more enlightening graph than last time. Take your dodge% from gear and buffs on the x-axis and parry% from the same on the y-axis. If that point is below the line, parry rating is better than dodge rating. If it is not, dodge rating is better than parry rating.

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