Intuitive Theories of Mind: A Rational Approach to False Belief

Noah D. Goodman1, Chris L. Baker1, Elizabeth Baraff Bonawitz1, Vikash K. Mansinghka1

Alison Gopnik2, Henry Wellman3, Laura Schulz1, Joshua B. Tenenbaum1

1Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology,2University of California, Berkeley, 3University of Michigan

Abstract

We propose a rational analysis of children’s false beliefreasoning. Our analysis realizes a continuous, evidence-driven transition between two causal Bayesian modelsof false belief. Both models support prediction and ex-planation; however, one model is less complex whilethe other has greater explanatory resources. Becauseof this explanatory asymmetry, unexpected outcomesweigh more heavily against the simpler model. We testthis account empirically by showing children the stan-dard outcome of the false belief task and a novel “psy-chic” outcome. As expected, we find children whoseexplanations and predictions are consistent with eachmodel, and an interaction between prediction and ex-planation. Critically, we find unexpected outcomes onlyinduce children to move from predictions consistent withthe simpler model to those consistent with the morecomplex one, never the reverse.

In everyday life we often attribute unobservable men-tal states to one another, and use them to predict and ex-plain each others’ actions. Indeed, reasoning about otherpeople’s mental states, such as beliefs, desires, and emo-tions, is one of our main preoccupations. These abilitieshave been called theory of mind (Premack and Woodruff,1978); theory of mind has become one of the most well-studied, and contentious, areas in modern psychology.In particular, much research has focused on the phe-nomenon of false belief: the ability to infer that othershold beliefs which differ from the (perceived) state of theworld. An often-used assay of this ability is the standardfalse belief task (Wimmer and Perner, 1983): the sub-ject sees Sally place her toy in a basket, then go out toplay. In Sally’s absence her toy is moved to a box (caus-ing her belief about the toy’s location to be false). Thesubject is then asked to predict Sally’s action: “whenSally comes back in, where will she look for her toy?”Many authors have reported that performance on thistask undergoes a developmental transition in the thirdor fourth year of life, from below-chance to above-chanceperformance (see Wellman et al. (2001) for a review andmeta-analysis, though see also Onishi and Baillargeon(2005)).

It has been suggested (Carey, 1985) that domainknowledge, such as theory of mind, takes the form ofintuitive theories, or coherent “systems of interrelatedconcepts that generate predictions and explanations inparticular domains of experience” (Murphy, 1993). Thisviewpoint leads to an interpretation of the false belieftransition as a revision of the child’s intuitive theory

from a copy theorist (CT) position about beliefs (i.e. be-liefs are always consistent with the world) to a perspectivetheorist (PT) position (i.e. beliefs are mediated by per-spective, and can be false). Perhaps the most strikingcomparison between these two theories is the asymmetryin their explanatory resources: the PT theory can makefalse belief predictions and explanations while the CTtheory cannot.

Another influential thread of research has supportedthe idea that human behavior is approximately ratio-nal within its natural context (Anderson, 1990). Withincognitive development both strong and weak versions ofthis thesis are possible. On the strong interpretationchildren respond and learn rationally throughout devel-opment; developmental stages can thus be analyzed asindividually optimal, in context, and collectively as a ra-tional progression driven by experience. On the weakreading it is only the final, mature, state which can beexpected to be rational. The contrast between these in-terpretations has played out vividly in research on falsebelief (cf. Leslie, 1994; Gopnik and Wellman, 1992).

Empirically, the false belief transition is slow: chil-dren do not immediately achieve false belief understand-ing when exposed to evidence prima facie incompatiblewith the CT position (Amsterlaw and Wellman, in press;Slaughter and Gopnik, 1996). This presents a puzzle forstrong rationality: how could it be rational to maintaina CT position about beliefs in the face of prediction fail-ures? That is, why would one ever accept a theory withfewer explanatory resources, unless perhaps, there is adrawback to the greater flexibility of the alternative the-ory? Indeed, intuition suggests that greater explanatoryability must come at the cost of greater complexity and,by Occam’s razor, it should thus require additional evi-dence to accept the richer theory.

However, it is difficult from an informal descriptionto know how explanatory resources and complexity dif-fer between these theories, and how these factors shouldinteract with evidence. Gopnik et al. (2004) have sug-gested that intuitive theories may be represented ascausal Bayesian networks (Pearl, 2000); we use thisframework to specify two models1 of false belief. Byapplying Bayesian methods we investigate the rationaltransition between these models, balancing explanatory

1Our formal analysis takes place at Marr’s computationallevel of modeling (Marr, 1982), that is, we describe the com-petencies, but not the algorithms (or processes), of cognition.

resources against complexity, and illuminate the aboverevision puzzle. To probe these ideas experimentally weinvestigate children’s predictions and explanations, incases when these predictions succeed and when they fail:the false belief task with the standard outcome (surpris-ing to CTs), and a novel “psychic” outcome (surprisingto PTs). We present only the apparatus necessary fora first investigation, leaving important elaborations forfuture work.

Formal ModelsIn the standard false belief task, described earlier, thestory begins with Sally putting her toy in the basket.As the story continues there are only three (observable)variables that have multiple outcomes: the final positionof the toy, Sally’s visual access to the final position (i.e.whether the door of the basket and box are open), andSally’s action upon re-entering the room. Thus we havethe variables World, Visual Access, and Action availableto our models (see Table 1 for descriptions). In addi-tion, there are two unobservable mental state variables:Sally’s belief about the location of her toy, Belief, andher Desire. We simplify the, presumably sophisticated,sub-theory of goals and desires (see Baker et al., in press)by collapsing desires into one variable, which indicateswhether Sally’s primary desire is her toy. (Formally, wemarginalize out all other variables in this sub-theory.)

V

W B D

A

ε(a) (b)

V

W B D

A

εγ

Figure 1: The dependency graphs of our Bayesian NetworkModels: (a) CT model, (b) PT model. Variables abbreviatedby their first letter (see Table 1).

To specify the relationships between these variableswe fix their joint distribution by giving a causal Bayesiannetwork. The pattern of conditional dependencies, givenby the directed graphs in Fig. 1, codifies the intuitionthat action is determined by beliefs and desires, and thatbelief is affected by the state of the world. In the PTmodel belief also depends on access2.

The conditional dependencies are parameterized bythe conditional probabilities given in Table 1. The con-ditional probability table for action describes a simplecase of the rational agent assumption: a person will actrationally, given her beliefs, to achieve her desires. Inthis case, if Sally wants her toy she will go to the loca-tion she believes it to be in, otherwise she goes to eitherlocation with equal probability (surely a simplification,but sufficient for present purposes). The variable De-sire has prior probability 1 − ε, which will be large for

2This is a simplification: we model how belief content de-pends on access, but it is likely that access mediates knowl-edge (vs. ignorance) even in the earlier theory.

desirable objects (such as a toy).For the CT model, Belief is constrained to equal

World. This is also true for the PT model when Vi-sual Access is present, but without access Sally main-tains her original belief, Belief = 0, with probability1 − γ. The parameter γ represents all the reasons, out-side of the story, that Sally might change her mind: hersister might tell her the toy has moved, she may haveE.S.P., she may forget that she actually left her toy inthe basket....

We assume asymmetric-beta priors on ε and γ. Inthe example simulations described below (Figures 2 and3) the hyper-parameters were set to β(1, 10) for ε, in-dicating that Sally probably wants her toy, and β(1, 5)for γ, indicating that she is unlikely to change her belief(lacking access). The relative magnitude of the two pa-rameters determines whether it is more likely that Sallywants something other than her toy, or that she changesher belief – we have chosen the latter (because standardfalse belief tasks emphasize that Sally wants her toy).Otherwise, the qualitative results described below arequite insensitive to the values of these parameters.

PredictionHaving represented our models as probability distribu-tions, rational predictive use is now prescribed by thealgebra of probabilities: conditioned on observation ofsome subset of the variables, a posterior distribution isdetermined that predicts values for the remaining vari-ables. These models are causal theories: they also sup-port predictions of the outcome of interventions, via thecausal do operator.)

Take the example in which Sally’s toy has been moved,but Sally doesn’t have visual access to this new loca-tion (see schematic Fig. 2(a)). There are two possibleoutcomes: Sally may look in the original location (thebasket), or the new location (the box). We may pre-dict the probability of each outcome by marginalizingthe unobserved variables. Fig. 2(b) shows that the twomodels make opposite predictions. We see that the CTmodel “fails” the false belief test by predicting that Sallywill look in the new (true) location, while the PT model“passes” by predicting the original location. The sur-prising outcome cases differ for the two models (lookingin the original location for CT, looking in the new loca-tion for PT). Note that while the surprising outcome isnot impossible in either model, it is far less likely in theCT model (as evident from Fig. 2(b)). That is, there isan explanatory asymmetry: prima facie equivalent un-expected outcomes weigh more heavily against the CTmodel than the PT model.

Theory RevisionStrong rationality requires an agent to balance the avail-able intuitive theories against each other. How should atheory-user combine, or select, possible theories of a do-main, given the body of her experience? Fortunately, thealgebra of Bayesian probability continues to prescribe ra-tional use when there are competing models: the degreeof belief in each model is its posterior probability given

Variable Description StatesWorld (W ) Location of the toy. 0: Original location, 1: New location.Access (V ) Could Sally see the toy moved? 0: No, 1: Yes.Action (A) Where Sally looks for her toy. 0: Original location, 1: New location.Belief (B) Where Sally thinks the toy is. 0: Original location, 1: New location.Desire (D) Sally’s primary desire. 1: To find the toy, 0: Anything else.

P (A = 1|B,D) B D0 0 11 1 1

0.5 0 00.5 1 0

PCT(B = 1|W ) W0 01 1

PPT(B = 1|W,V ) W V0 0 11 1 1γ 0 0γ 1 0

Table 1: The random variables and probability distribution tables for our models.

CT PT0

0.2

0.4

0.6

0.8

1P(A=0|W,V)

CT PT0

0.2

0.4

0.6

0.8

1P(A=1|W,V)

Originallocation

Originallocation

Newlocation

Newlocation

Initial Final

Protagonistlooks

(a) (b) (c) (d)

CT PT0

0.5

1

1.5

2

2.5Surprise of posterior mode

D

Eγ BDB

CT PT0

0.5

1

1.5

2Surprise of posterior mode

D

Eγ B

DB

Stan

dar

d o

utc

om

ePs

ych

ic o

utc

om

e

CT PT0

0.2

0.4

0.6

0.8

1D:0B:1

D:1B:0Eγ:0

D:0B:0Eγ:0

D:0B:1Eγ:1

P(D,B|W,V,A) P(D,B,Eγ|W,V,A)

CT PT0

0.2

0.4

0.6

0.8

1

D:0B:1

D:1B:1

D:1B:1Eγ:1 D:0

B:0Eγ:0

D:0B:1Eγ:1

P(D,B|W,V,A) P(D,B,Eγ|W,V,A)Originallocation

Originallocation

Newlocation

Newlocation

Initial Final

Protagonistlooks

Figure 2: Comparing the Models: (a) Example situations, in both cases W=1, V =0, for the Standard outcome A=0, for thePsychic outcome A=1. (b) The predicted probability of each outcome. (c) Posterior probability of configurations of hiddenvariables, after observing the outcome. This indicates degree of belief in the corresponding complete explanation. (d) Surprisevalue of each variable in the modal configuration (computed as surprisal with respect to the predictive posterior).

previous experience. We may then write down a be-lief weight comparing belief in the PT model to the CTmodel:

WPT/CT = − log(P (PT|X)/P (CT|X)), (1)

where X represents experience in previous false beliefsettings. When WPT/CT is strongly negative the con-tribution from PT is negligible, and the agent behavesas though it is a pure CT. If evidence accumulates andshifts WPT/CT to be strongly positive the agent behavesas a PT. In Fig. 3 we plot WPT/CT evaluated on accu-mulating “epochs” of experience. Each epoch consists oftrials with (W,V,A) observed, but (D,B) unobserved.The trials in each (identical) epoch encode the assump-tions that visual access is usually available, and that,in instances without access, the protagonist often hasa correct belief anyway (e.g. to a child, his parents of-ten appear to have preternatural knowledge). (Specif-

ically, each epoch is twenty (W=1,V =1,A=1) trials,six (W=1,V =0,A=1) trials, and one (W=1,V =0,A=0)trial.) The expected transition from CT to PT does oc-cur under these assumptions. Since this rational revisiondepends on the particular character and statistics of ex-perience, a developmental account is incomplete withoutempirical research on the evidence available to childrenin everyday life.

How can we understand the delayed confirmation ofthe PT model? First, in the initial epoch, the CT modelis preferred due to the Bayesian Occam’s razor effect(Jefferys and Berger, 1992): the PT model has additionalcomplexity (the free parameter γ), which is penalized viathe posterior probability. However, the data themselvesare more likely under the PT model – because some ofthe data represent genuine false belief situations. Asdata accumulates the weight of this explanatory advan-tage eventually overcomes complexity and the PT model

Figure 3: The log-posterior odds ratio over data epochs,showing the false belief transition from CT to PT. (Parame-ters integrated numerically by grid approximation.)

becomes favored.When WPT/CT is close to threshold, inferences will be

mixed and far more sensitive to evidence3. In particular,any effect of the explanatory asymmetry noted aboveshould be particularly prevalent in this period. When farfrom threshold, predictions in the situation of Fig. 2(a)will be largely consistent, but close to threshold theymay become mixed, and such variability will be greaterbelow threshold than above.

ExplanationAs the above discussion emphasizes, an important func-tion of intuitive theories is to explain observations byhypothesizing states of unobserved variables. To modelthis explanatory competence we first recast our modelsinto a deterministic explicit-noise form, by introducingadditional variables, in order that observations will fol-low necessarily from unobserved variables. Explanationcan then be described as inference of a complete expla-nation – a setting of all variables – and communicationof a portion of this complete explanation, the explanans.

Our PT model becomes deterministic if we introducean External Information (or ‘alternate access’) variable,Eγ (with prior probability γ), to explicitly representevents which cause changes in belief (in the absence ofaccess). (For completeness, an additional variable thatdetermines the object of alternate desires could be in-cluded; for technical reasons this variable has minimaleffect, and has been omitted for clarity.) Explanatoryinference is now dictated by the posterior distribution,conditioned on observations: the degree of belief in eachcomplete explanation is given by its posterior probabil-ity (e.g. Fig. 2(c)). (See Halpern and Pearl (2001) for arelated approach.) Once a complete explanation is cho-sen, a partial account of the explanans can be given byappealing to the principle of surprise: a good explananswill address the ways in which an explanation is surpris-ing4.

We can now characterize the explanatory asymmetrybetween the two models in more detail. Comparing thedependency structure of the two models, we see that ac-

3One expects the details of a process-level account to beespecially critical near the threshold.

4Surprise may be formalized by the information-theoreticsurprisal of a value with respect to some reference distribu-tion, such as the predictive posterior.

cess, alternate access (including external information),and belief (independent from the world) are causally rel-evant variables only for the PT model. We thus expectPT-theorists to appeal to belief and access more thanCT-theorists, while CT-theorists will appeal primarilyto desire. Indeed, to explain a surprising experience theCT model can only infer an alternate desire – Sally wentback to the basket because she wanted the basket (notthe toy). The PT model has additional explanatory re-sources in the sense that it can also appeal to access, par-ticularly alternate forms of access, to explain unexpectedoutcomes – Sally went to the box because someone toldher that her toy was there. Fig. 2(d) shows that thesealternate desire and alternate access variables are indeedthe surprising aspects of the most relevant complete ex-planations. From the principle of surprise we may thenpredict an interaction between theory and explanationtype: in the surprising outcome cases CT-theorists willappeal to alternate desires, while PT-theorists will ap-peal to alternate access.

Children’s Predictions and Explanations

If children are using intuitive theories of mind, as de-scribed here, then several model predictions should hold.First, there will be an evidence-driven false belief tran-sition, and there will be a group of children near thethreshold of this transition who exhibit decreased coher-ence in prediction and explanation. Because of the ex-planatory asymmetry between models, we predict thatsurprising outcomes will have greater weight before thetransition than after. In particular, we expect some chil-dren who begin with CT predictions to switch to PT pre-dictions when they encounter surprising evidence, butfew children to switch from PT predictions to CT pre-dictions when they encounter surprising evidence, andfew children in any condition to switch when given con-sistent evidence.

Among children who are relatively far from threshold,and thus provide consistent predictions, there should bean interaction between prediction type and explanations:PT-predictors should generate more belief and access re-sponses, while CT-predictors should generate more de-sire responses. Accordingly, we investigated the predic-tions and explanations that children generated when pre-sented with the two possible outcomes to the standardfalse belief story. This is an extension to the prediction-explanation paradigm used by several authors to inves-tigate false belief (e.g. Bartsch and Wellman, 1989).

Participants Forty-nine children (R=3;0-4;11,M=3;11) were tested in a quiet corner of an interactivescience exhibit at a local museum. Parents were visibleto the child, but were instructed not to interact withthe child during the study.

Materials and Procedure Three picture books werecreated. The first book presented a guessing game un-related to the false-belief task and was used to familiar-ize the child with the experimenter and with generatingguesses. The other two books, Standard and Psychic, fol-lowed the standard Sally-Anne style narrative, with car-

toon pictures depicting the events throughout the book.These stories are equivalent to the situations of Fig. 2(a).

Standard outcome book: Sally was shown hiding herteddy-bear in a basket before going outside. Childrenwere asked to point out where the teddy-bear was be-ing hidden. Then a mischievous character, Alex, wasshown moving the teddy-bear from the basket to thebox while Sally was away. As a memory check childrenwere asked where the teddy-bear was moved. On thethird page, Sally starts to come back into the room, andthe children were asked, “Here comes Sally. Where doyou think Sally is first going to go to get her toy?” Afterthe children responded, the next scene depicted Sally go-ing to the basket to get her toy. The children were thenprompted with, “Sally went to look for her bear in thebasket. Sally’s bear is really in the box. But Sally islooking for it in the basket ! Why is she looking there?”The children were given the chance to respond. If theywere unable to provide an explanation or provided unin-formative information, the experimenter repeated, “Yes,but she’s looking for it way over here. What happened?”

Psychic outcome book: The procedure and dialog wereessentially identical to the Standard outcome book, ex-cept the main character searched for his missing item inthe location to which the item was moved, not where itwas left. There were also superficial differences involvingdifferent characters (Billy & Anne), different objects (acookie), and different locations (a drawer and a cabinet).Predictions and explanations were elicited as before. Theorder of the two test books was counter-balanced.Results and Discussion Four children failed thememory test, and were excluded from further analysis.Based on the childrens responses to the prediction ques-tion in each book, we found three groups of children: 15children who predicted the new location in both books(CT-predictors), 20 children who predicted the origi-nal location in both books (PT-predictors), and 10 chil-dren who changed their prediction based on the surpris-ing evidence (Mixed-predictors). Critically, and as pre-dicted by the explanatory asymmetry between the mod-els, children only switched predictions after surprisingevidence, and no children moved from the PT-consistentprediction to the CT-consistent prediction. Thus, ofthe children who initially made a CT-prediction, sig-nificantly more of those who received surprising evi-dence (for them the standard outcome) switched thanof those who received confirming evidence (p < 0.01,χ2 = 7.84), and of the children who received surpris-ing evidence significantly more who initially made CT-predictions switched than of those who initially madePT-predictions (p < 0.01, χ2 = 8.60). This order ef-fect cannot be explained as a simple response to pre-diction failure, because the PT-predictors showed noincreased tendency to switch when they received sur-prising evidence (for them the Psychic condition). Onemight worry that only the younger children were flus-tered enough by a wrong prediction to begin randomguessing. This would imply that older children shouldbe less likely to switch predictions than younger chil-dren. In fact, comparing the ages of the CT-predictors

(M=3;7) to those of the Mixed-predictors (M=3;10) re-vealed that the Mixed-predictors were significantly olderthan the CT-predictors (p < 0.05, t = 2.06). This sug-gests that this pattern of mixed predictions is indicativeof children who are close to the false belief transition.

Explanations were coded for the mention and value,if any, of each variable as in Table 1. (For the remain-der, we have combined External Information with Ac-cess for clarity.) For example, one (PT-predictor) childexplained the Psychic outcome by “I think he heard hissister going over there,” and this was coded as Access=1.Another (CT-predictor) child explained the Standardoutcome by “well, that’s where she wants to look,” whichwas coded Desire=0. Responses were scored by twocoders, one who was blind to the group type for eachchild and to the formal model; inconsistencies were re-solved by discussion. There were no significant differ-ences between groups in references to observed variables(Initial World, Final World, Action).

Alternate DesireAlternate Access

020406080

100

PT CT

Mix PT CT

Mix PT CT

Mix

Standard

Psychic

Belief Access Desire

(a) (b)

0

5

10

15

20

PT CT

Figure 4: Summary of explanation data: (a) Portion of re-sponses for each prediction group and condition referencingBelief, Access, and Desire. (b) Portion of responses assertingalternate values of Desire and Access, by prediction group.

As predicted, more CT-predictors than PT-predictorsgave Desire explanations (p < 0.025, χ2 = 5.60, acrossconditions), and more PT-predictors than CT-predictorsgave Belief or Access explanations (p < 0.025, χ2 = 6.60,across conditions). The Mixed-predictors gave explana-tions that were quite similar to the CT-predictors: theirreferences to both Desire and Belief+Access were signif-icantly different than the PT-predictors (both p < 0.01by χ2), but not the CT-predictors. Children’s explana-tions are summarized in Fig. 4.

There is no way to know the chance level for explana-tions. However, consider that for an explanation to becoded as referring to alternate access or alternate desires,children had to spontaneously invent and report detailsoutside the story (e.g. “he heard his sister move it”, or“she wanted to move the basket”). It is thus suggestivethat three of twenty PT-predictors gave alternate accessexplanations, and two of fifteen CT-predictors gave al-ternate desire explanations, in the respective surprisingoutcome conditions. Further, as predicted by our formalanalysis, there is a significant interaction between predic-tion group (CT vs. PT) and type of the alternate expla-nations that were offered (desire vs. access) (p < 0.05 by2×2 mixed ANOVA, F (1, 136) = 5.04). The interactionremains significant when restricting to the surprising

outcome conditions (p < 0.05 by 2×2 mixed ANOVA,F (1, 66) = 5.31). Interestingly, the Mixed-predictors of-fer significantly more alternate explanations (access ordesire) than the non-mixed (PT and CT) predictors(p < 0.025, χ2 = 5.02).

Conclusion

The history of developmental psychology has been filledwith tension between the view that children are incom-plete minds biding their time until full maturation, andthe view that they are rational agents bootstrappingtheir way to an understanding of the world. The no-tion that children are strongly rational is alluring, as itwould provide a uniform principle from which to under-stand development.

We have outlined a computational account of theory ofmind as applied to the false belief task. This frameworkrealizes false belief reasoning as rational use and revisionof intuitive theory. Few formal models have been previ-ously presented to account for false belief, and, to thebest of our knowledge, none of these other models givesa strongly rational account. The CRIBB model of Wahland Spada (2000), for instance, approaches failures offalse belief as the result of limited processing capability.

Two of the primary advantages of formal models havebeen illustrated here. First, added precision can illu-minate theoretical problems that resist simple solution.Indeed, our computational account sheds some light onthe puzzle of rational theory revision by bringing thetools of Bayesian analysis to bear on the tradeoff be-tween explanatory resources and complexity. Second, amodel can suggest novel experimental avenues. Consid-eration of the formal structure of our models, especiallythe explanatory asymmetry between them, suggested de-signing an outcome condition which would be surprisingto children who passed the false belief test – the novelPsychic outcome condition of our experiment. This con-dition then provided the crucial contrast needed to un-derstand the interaction between theory and explana-tion, and to detect the outcome-order effect in predic-tions. These in turn suggest further experimental andtheoretical avenues, such as a training study to test oursuggestion that certain mixed predictions are a signatureof children very near to the false belief transition.

The present account of the false belief transition is in-complete in important ways. After all, our agent hadonly to choose the best of two known models. This begsan understanding of the dynamics of rational revisionnear threshold and when the space of possible modelsis far larger. Further, a single formal model ought ulti-mately to be applicable to many false belief tasks, andto reasoning about mental states more generally. Severalcomponents seem necessary to extend a particular the-ory of mind into such a framework theory: a richer rep-resentation for the propositional content and attitudes inthese tasks, extension of the implicit quantifier over trialsto one over situations and people, and a broader view ofthe probability distributions relating mental state vari-ables. Each of these is an important direction for futureresearch.

AcknowledgmentsThanks to the Boston Museum of Science, participants ofthe McDonnell Workshops (2005), Rebecca Saxe, Tania Lom-brozo, and Tamar Kushnir. This research was supported bythe James S. McDonnell Foundation Causal Learning Collab-orative Initiative.

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