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REVIEW 3 major objections 6 minor 34 references

Stephenson 2 DFK 52: Discovery of an exotic red supergiant in the massive stellar cluster RSGC2

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

Pith's one-line read DFK 52, a seemingly ordinary red supergiant in the cluster RSGC2, is surrounded by a 50,000-au circumstellar outflow, the largest seen around an evolved star, and its low luminosity points to a mass-ejection mechanism unlike any known…

desk verdict Solid ALMA detection of an unprecedentedly large circumstellar envelope around DFK 52, but the exotic interpretation hinges on an assumed cluster distance and a model that fits only 15% of the flux. read the letter →

arxiv 2507.11609 v1 pith:MAFOAOAT submitted 2025-07-15 astro-ph.GA astro-ph.SR

classification astro-ph.GAastro-ph.SR
keywords redsupergiantcircumstellarenvelopemasslossALMARSGC2Stephenson2supernovaprogenitorradiativetransfer
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper reports that DFK 52, a red supergiant in the massive cluster RSGC2, is surrounded by a circumstellar outflow that extends up to 50,000 au, roughly four times farther than the largest outflows around other known extreme red supergiants. The outflow is asymmetric and complex, with detached dust clumps, arcs, and a bar-like equatorial structure seen in carbon monoxide that expands at 27 km s$^{-1}$, alongside a slower spherical wind. The authors model this as a dramatic mass-loss episode about 4,000 years ago that has since weakened to a gentler wind, and they estimate that $0.1$ to $1$ $M_\odot$ of gas and dust now reside in the extended envelope. Because DFK 52 is roughly ten times less luminous than the known extreme red supergiants, they conclude that the ejection mechanism may be unique and requires an energy source beyond ordinary radiation pressure.

What carries the argument

The two-component radiative-transfer model built with the LIME code is the central analytical object. It combines a spherically symmetric present-day wind (mass-loss rate $3\times10^{-6}\,M_\odot\,\mathrm{yr}^{-1}$, expansion velocity 10 km s$^{-1}$) with a detached equatorial density enhancement, an edge-on torus parameterized as a two-dimensional Gaussian in radius and height, holding $0.05\,M_\odot$ centered at 23,000 au and expanding at 27 km s$^{-1}$. Fitting this model to the position-velocity diagram along the bar yields the 4,000-year age of the ejection; scaling the same component to the other extended structures gives roughly $0.35\,M_\odot$ of detached material, and a momentum-budget analysis using the same geometry supplies the argument that radiation pressure alone cannot explain the event.

What would settle it

Measure DFK 52's distance directly, for example with a parallax or an improved cluster distance with uncertainty well below 20%. At 1.5 kpc the outflow radius would shrink to about 13,000 au, placing the object on the scale of VY CMa and undermining the unprecedented-size claim, while a distance larger than 5.8 kpc would make the envelope even more extreme; the paper itself states that such a measurement is needed.

Watch

Extended reading notes

Core claim

On its own terms, the paper's claim is that DFK 52 is an exotic red supergiant whose circumstellar environment is unlike any previously observed. ALMA 1.3 mm images show continuum emission reaching a radius of about 50,000 au (at the assumed 5.8 kpc distance) and $^{12}$CO $J=2-1$ emission reaching about 45,000 au, with no central continuum peak at the stellar position. The morphology is organized into arcs, loops, a detached bar-like structure that appears in position-velocity space as an edge-on equatorial ring expanding at 27 km s$^{-1}$, and a compact slow central wind; the low $^{12}$CO/$^{13}$CO ratio of about 5 supports a stellar, CNO-processed origin for the material. The paper interprets the structure as a massive ejection event about 4,000 years ago, followed by a return to slower, more symmetric mass loss, and it argues that neither the current radiation pressure nor standard single-star evolution can easily produce the inferred momentum.

Load-bearing premise

The load-bearing premise is that DFK 52 is a member of RSGC2 and therefore lies at the cluster distance of 5.8 kpc; every physical size, mass, luminosity, and comparison with other extreme red supergiants is scaled by that distance.

Editorial extensions

If this is right

  • DFK 52 demonstrates that a red supergiant can undergo a short, violent mass-loss episode and then return to a much slower wind, so dramatic mass loss need not mark the end of the star's life.
  • The $0.1$ to $1$ $M_\odot$ of cold material at radii beyond $10^{17}$ cm will surround the star when it explodes, so DFK 52 is a concrete example of the dense circumstellar medium invoked to shape some Type II supernova light curves.
  • If the ejected ring was produced by a superwind, the episode ended without a supernova, constraining how superwind phases relate to core collapse.
  • If it was instead produced by a companion interaction or merger, DFK 52 may be a direct progenitor candidate for ultra-stripped supernovae or a gravitational-wave compact binary.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: a binary or merger origin could be tested by looking for a surviving companion or merger remnant through high-angular-resolution imaging or radial-velocity monitoring of DFK 52.
  • The paper's own model reproduces only 15% of the total CO flux, so a full 3D non-LTE model of all channel maps would test whether the mass estimate of $0.1$ to $1$ $M_\odot$ holds or needs revision.
  • The unusually cold, detached dust and double-peaked SED suggest that episodic mass loss can occur in lower-luminosity RSGs; a millimetre survey of other cluster members could reveal whether DFK 52 is truly unique or the first recognized member of a class.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper presents ALMA 1.3 mm continuum and CO/SiO line observations of the red supergiant DFK 52 in the cluster RSGC2. The authors detect an extended, complex circumstellar envelope reaching ~8 arcsec (claimed ~50,000 au at the assumed 5.8 kpc cluster distance), with a bar-like CO structure whose position-velocity diagram suggests an edge-on detached equatorial component moving at ±27 km/s. A two-component LIME radiative transfer model — a slow spherical wind (3e-6 Msun/yr, 10 km/s) plus a detached equatorial density enhancement (0.05 Msun, 27 km/s, centered at 23,000 au) — reproduces the PV diagram along one position angle but only 15% of the total CO flux. The authors infer a dramatic mass-loss event about 4000 years ago, estimate a total outflow mass of 0.1–1 Msun, and argue that the size and low luminosity make DFK 52 unique among extreme red supergiants.

Significance. If the distance and membership assumption holds, this is a potentially important discovery of an RSG with an outflow roughly four times larger than those of VY CMa and NML Cyg and with a much lower luminosity, which would challenge current understanding of RSG mass loss and supernova progenitor scenarios. The observations are of high quality, reduced with standard ALMA procedures, and the kinematic evidence for a detached fast component is visually compelling. The paper is commendably transparent about the two main caveats: the dependence of all physical scales on the assumed cluster distance (Appendix D) and the fact that the radiative transfer model reproduces only a small fraction of the CO flux (§3.2). The low 12CO/13CO ratio supporting a stellar origin is a nice check. However, because the 'exotic' classification and the inferred mass-loss history both rest on these caveats, the quantitative claims need to be presented with explicit systematic uncertainties.

major comments (3)
  1. [§3.1, Appendix D] All physical quantities that make DFK 52 exotic — the ~50,000 au envelope radius, the 4000 yr ejection age, the 0.1–1 Msun envelope mass, the 2×10^4 Lsun luminosity, and the factor 3–4 size comparison with VY CMa and NML Cyg — are directly proportional to the assumed cluster distance d = 5.8 kpc from Davies et al. (2007). The paper itself notes in Appendix D that the quoted distance uncertainty alone changes the 8 arcsec extent from ~40,000 au (d=5.04 kpc) to ~62,000 au (d=7.71 kpc), and that a foreground star at d=2 kpc would reduce the envelope to ~16,000 au and the luminosity to ~2400 Lsun, which would challenge the RSG classification. The radial-velocity argument (v_lsr = 109 km/s vs. <30 km/s expected for d<2 kpc) makes such a foreground interloper improbable, but not impossible. Because the central claim of an unprecedented outflow rests on this distance, I request that the paper present the key physical quantities explicitly as functions of d (e.g., R = 8 arcsec × d), and that the abstract and conclusions be worded conditionally on the membership assumption. Stronger membership confirmation (e.g., Gaia astrometry or a cluster parallax) would strengthen the claim.
  2. [§3.2, Table B.1, Appendix B] The inferred mass-loss event — an equatorial component with M=0.05 Msun ejected ~4000 yr ago at 27 km/s — is derived from the LIME model parameters (r_EDE = 23,000 au, v_EDE = 27 km/s, M_EDE = 0.05 Msun) that were chosen to reproduce the PV diagram along PA=143°. The age follows essentially as r_EDE/v_EDE, so it is not an independent measurement. The model reproduces only 15% of the total CO flux and requires an unmodeled azimuthal density asymmetry to account for the missing blueshifted torus emission. The paper should explicitly label the 4000-yr mass-loss event as a model-dependent interpretation and should provide a sensitivity test showing how the derived mass and age vary when the model's free parameters (opening angle, radial width, abundance, temperature) are varied within plausible ranges.
  3. [§3.1, §5] The abstract's total outflow mass of 0.1–1 Msun rests on two extrapolations: the continuum dust mass (Md ~ 6.6×10^-3 Msun at T=50 K, beta=0.9, gas-to-dust=200) and the CO model scaled to the unmodeled flux (M~0.35 Msun assuming common excitation conditions). Both estimates assume optically thin emission and adopt a single dust temperature and fixed gas-to-dust ratio; the systematic uncertainties from these choices are not propagated. Since the mass range is one of the headline results, the paper should provide a conservative systematic error budget or explicitly state that the quoted range is an order-of-magnitude estimate rather than a measured value.
minor comments (6)
  1. [§3.1] The sentence 'Together, these structures account for ~15% of the total flux' is ambiguous: it is not clear whether 'these structures' refers to the compact components A/B/C or to the extended low-brightness component. Please rephrase.
  2. [Abstract and §3.1] The abstract states 'up to 50,000 au in radius,' while the maximum angular extent is 8 arcsec (46,000 au at 5.8 kpc) for the continuum and 7.5 arcsec for CO. Please specify that 50,000 au is a rounded value that includes the distance uncertainty, or quote the angular sizes as the primary measurement.
  3. [§4] The comparison of the envelope extent with 'the smallest observed galactic clumps (0.1 pc; Urquhart et al. 2021)' is misleading because 0.1 pc = 20,600 au, about a factor of two smaller than the claimed 45,000–50,000 au extent. Please quantify the comparison in the same units.
  4. [Appendix B] The notation for the temperature power law is inconsistent: the text says the log-slope alpha = 1, while Table B.1 lists Tkin = T⋆ (r/R⋆)^(−α) without giving a numeric value for alpha. Please make the notation uniform.
  5. [§3.2] The statement that the CO extent is 'roughly four times larger than the farthest components observed in this line for VY CMa... and NML Cyg' should specify the assumed distances of VY CMa and NML Cyg and confirm that the comparison is in physical size, not angular size.
  6. [§2] There is a typo in 'di fferent' in the Observations section.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the model parameters are data-constrained and the inferred 4000 yr age is a derived timescale, not a fitted prediction.

full rationale

The claimed derivation chain from ALMA data to physical conclusions is not circular. The 50,000 au extent is the observed 8'' continuum radius converted at the assumed cluster distance (5.8 kpc; Appendix D), and the paper explicitly flags that distance is an assumption needing confirmation. The 4000 yr ejection age is computed as r_EDE/v_EDE from the fitted LIME model (r_EDE = 23,000 au, v_EDE = 27 km/s; Table B.1), but these parameters are constrained by the observed PV diagram and line profiles; the age is a derived timescale, not a separately fitted quantity presented as an independent prediction. The paper also states the RT model reproduces only 15% of the total CO flux, so the mass-loss history is presented as a suggestion, not an overclaimed prediction. The only overlapping-author citation (De Beck et al. 2025) is used as a comparison/method reference for NML Cyg and is not load-bearing. No equation is equivalent to its input by construction.

Assumptions & free parameters 13 free parameters · 6 assumptions · 0 invented entities

The central interpretation rests on a cluster distance, an optically thin cold dust assumption, and a set of LIME model parameters fitted to the observed morphology. The paper is transparent about these, but they are not independently derived. No new physical entities are introduced; the equatorial density enhancement is a parameterized density structure, not a new particle or force.

free parameters (13)
  • Equatorial density enhancement mass (M_EDE) = 0.05 M_sun
    Fitted in the LIME radiative transfer model to match the brightness of the barred CO structure; directly sets the inferred ejecta mass.
  • Equatorial component expansion velocity (v_EDE) = 27 km/s
    Chosen to reproduce the outer velocity extent of the PV diagram; with r_EDE it fixes the 4000-year age.
  • Equatorial component center radius (r_EDE) = 23000 au (4 arcsec)
    Set to the observed offset of the barred structure; a key input for the age and size claims.
  • Equatorial component radial width (sigma_EDE) = 6000 au
    Adopted Gaussian width in the model, chosen to fit the radial surface brightness profile.
  • Equatorial opening angle (theta_EDE) = 10 deg
    Assumed to match the vertical thickness of the bar and the radiation-interception fraction, not directly measured.
  • Slow spherical wind mass-loss rate (Mdot_sph) = 3e-6 M_sun/yr
    Fitted to compact CO emission; consistent with the earlier SED-based estimate, but not an independent prediction.
  • Slow wind expansion velocity (v_sph) = 10 km/s
    Chosen to match the central CO line width near the stellar position.
  • 12CO abundance relative to H2 = 2e-4
    Assumed in the LIME model; all molecular masses scale linearly with this value.
  • Minimum kinetic temperature (Tmin) = 15 K
    Ad hoc floor introduced because the adopted power-law temperature reaches unphysically low values at the torus radius.
  • Dust temperature of the cold envelope = 50 K
    Inferred from the far-IR SED peak and used to convert the ALMA continuum flux into dust mass; a factor of two error changes masses by a similar factor.
  • Dust emissivity index (beta) = 0.9
    Adopted from O'Gorman et al. (2015) and De Beck et al. (2025); affects dust mass estimates.
  • Gas-to-dust ratio = 200
    Adopted from Humphreys et al. (2020) to convert dust mass to total envelope mass, giving the 0.1-1 M_sun range.
  • Cluster distance = 5.8 kpc
    Adopted from Davies et al. (2007); converts angular scales to au and determines luminosities and masses. Carries a large asymmetric uncertainty and is the biggest lever on the central claim.
assumptions (6)
  • domain assumption DFK 52 is a member of RSGC2 and lies at the cluster distance of 5.8 kpc (Appendix D; Section 3.1).
    All physical sizes, luminosities, masses, and the comparison with VY CMa and NML Cyg scale with this distance. The paper argues for membership from velocity and position but explicitly requests a more accurate distance measurement.
  • domain assumption The detected extended CO and continuum emission is circumstellar rather than interstellar, outside the explicitly excluded ISM-contaminated channels.
    The 12CO/13CO ratio of about 5 and the absence of similar structures in other cluster RSGs favor a stellar origin, but the smooth extended continuum component resembles molecular cloud substructures and channels were excluded (Section 2, Section 4).
  • domain assumption The dust emission is optically thin, cold (about 50 K), and follows a fixed gas-to-dust ratio of 200.
    Used in Section 3.1 to derive dust masses and the 0.1-1 M_sun envelope mass from the 17.2 mJy continuum flux; supported only by a single spectral index extrapolation and prior SED fits.
  • ad hoc to paper The LIME two-component model parameters are a valid representation of the dominant physical structures.
    The model reproduces only 15 percent of the total CO flux and is fit along one position angle; scaling to the total mass assumes excitation conditions are common across the many extended structures (Section 3.2, Appendix B).
  • domain assumption Constant radial expansion since ejection.
    The 4000-year age and the pre-SN density comparison in Section 4 use r_EDE/v_EDE with no account of acceleration, deceleration, or interaction with the surrounding medium.
  • domain assumption The equatorial and spherical components do not interact radiatively.
    Appendix B states the components do not overlap spatially at any velocity, so their populations and optical depths are computed separately, simplifying the radiative transfer calculation.

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Pith. "Pith review of Stephenson 2 DFK 52: Discovery of an exotic red supergiant in the massive stellar cluster RSGC2." pith.science (2026). https://pith.science/paper/MAFOAOAT

@misc{pith2026250711609,
  author       = {Pith},
  title        = {Pith review of: Stephenson 2 DFK 52: Discovery of an exotic red supergiant in the massive stellar cluster RSGC2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MAFOAOAT}},
  note         = {Machine review of arXiv:2507.11609}
}
abstract

Atacama Large Millimeter/submillimeter Array (ALMA) observations at 1.3mm have recently revealed surprising complexity in the circumstellar environment of DFK 52, a red supergiant (RSG) located in the Stephenson 2 massive open cluster. We provide an initial characterisation of the star's mass-loss properties by studying its circumstellar emission in continuum, $^{12}$CO, $^{13}$CO, and SiO rotational lines. We find that DFK 52 is surrounded by an extremely large outflow (up to 50,000 au in radius) that shows complex morphologies in both its molecular and dust emission. The size of the circumstellar medium is unprecedented, even when compared with other known extreme RSGs, and its lower luminosity indicates that its mass ejection mechanism may be unique among this population. The molecular emission can be partially reproduced by a two-component model consisting of a fast (27 km/s) detached equatorial component with $M{\sim}0.05$ $M_{\odot}$ and a slow (10 km/s) spherical envelope with $\dot{M}\sim3\times10^{-6}$ $M_{\odot}$ yr$^{-1}$. This suggests that DFK 52 underwent a dramatic mass-loss event $\sim$4000 years ago, but has since transitioned into having a slower more symmetric mass loss. We conservatively estimate a total mass of $0.1-1$ $M_{\odot}$ in the complex extended regions of the outflow. The uncertain nature of the dramatic mass loss warrants extensive follow-up of this likely supernova progenitor.

Figures

Figures reproduced from arXiv: 2507.11609 by the authors.

Figure 1
Figure 1. 220 GHz continuum toward DFK 52. The stellar position (see Section 3.2) is marked with a blue cross, and the A, B, and C compo￾nents are labelled. The white contours are shown at [5, 10, 15]σ, and the dashed blue contour represents 3σ for the image residuals after smooth￾ing to a 2′′ resolution, tracing the low-brightness extended component. by a cold detached component. Using the same method as be￾fore, but for the… view at source ↗
Figure 2
Figure 2. Summary of molecular line emission observed toward DFK52. In the integrated CO map, the white contours denote the 220 GHz continuum emission at [5, 10, 15]σ, the blue dashed contours represent 3σ for the smoothed extended continuum component, the black cross marks the stellar position (as measured by the SiO emission peak), and the ALMA beam is displayed as a white filled ellipse in the lower right corner. The dashe… view at source ↗
Figure 3
Figure 3. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

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Works this paper leans on

34 extracted references · 30 canonical work pages

  1. [1]

    I., Sackmann , I

    Boothroyd , A. I., Sackmann , I. J., & Ahern , S. C. 1993, , 416, 762

  2. [2]

    & Hogerheijde , M

    Brinch , C. & Hogerheijde , M. R. 2010, , 523, A25

  3. [3]

    2001, , 377, 868

    Bujarrabal , V., Castro-Carrizo , A., Alcolea , J., & S \'a nchez Contreras , C. 2001, , 377, 868

  4. [4]

    F., Kudritzki , R.-P., et al

    Davies , B., Figer , D. F., Kudritzki , R.-P., et al. 2007, , 671, 781

  5. [5]

    2022, , 517, 1483

    Davies , B., Plez , B., & Petrault , M. 2022, , 517, 1483

  6. [6]

    De Beck , E., Andrews , H., Quintana-Lacaci , G., & Vlemmings , W. H. T. 2025, , 698, A179

  7. [7]

    A., Carter , M., Cheng , J., et al

    Ediss , G. A., Carter , M., Cheng , J., et al. 2004, in Fifteenth International Symposium on Space Terahertz Technology, ed. G. Narayanan , 181--188

  8. [8]

    F., Niccolini , G., Su \'a rez , O., et al

    G \'o mez , J. F., Niccolini , G., Su \'a rez , O., et al. 2018, , 480, 4991

Show all 34 references
  1. [9]

    & Negueruela , I

    Gonz \'a lez-Fern \'a ndez , C. & Negueruela , I. 2012, , 539, A100

  2. [10]

    M., Helmel , G., Jones , T

    Humphreys , R. M., Helmel , G., Jones , T. J., & Gordon , M. S. 2020, , 160, 145

  3. [11]

    Humphreys , R. M. & Jones , T. J. 2022, , 163, 103

  4. [12]

    M., Richards , A

    Humphreys , R. M., Richards , A. M. S., Davidson , K., et al. 2024, , 167, 94

  5. [13]

    R., Indebetouw , R., Brogan , C

    Hunter , T. R., Indebetouw , R., Brogan , C. L., et al. 2023, , 135, 074501

  6. [14]

    P., et al

    Justtanont , K., de Jong , T., Helmich , F. P., et al. 1996, , 315, L217

  7. [15]

    2019, , 627, A114

    Kami \'n ski , T. 2019, , 627, A114

  8. [16]

    2018, , 617, A129

    Kami \'n ski , T., Steffen , W., Tylenda , R., et al. 2018, , 617, A129

  9. [17]

    Khouri , T., Vlemmings , W. H. T., Tafoya , D., et al. 2021, Nature Astronomy, 6, 275

  10. [18]

    G., & Chruslinska , M

    Klencki , J., Nelemans , G., Istrate , A. G., & Chruslinska , M. 2021, , 645, A54

  11. [19]

    2011, , 417, 32

    Lagadec , E., Verhoelst , T., M \'e karnia , D., et al. 2011, , 417, 32

  12. [20]

    2024, , 688, A203

    Marton , G., Gezer , I., Madar \'a sz , M., et al. 2024, , 688, A203

  13. [21]

    J., Yoon , S.-C., Gr \"a fener , G., & Blinnikov , S

    Moriya , T. J., Yoon , S.-C., Gr \"a fener , G., & Blinnikov , S. I. 2017, , 469, L108

  14. [22]

    2012, , 547, A15

    Negueruela , I., Marco , A., Gonz \'a lez-Fern \'a ndez , C., et al. 2012, , 547, A15

  15. [23]

    O'Gorman , E., Vlemmings , W., Richards , A. M. S., et al. 2015, , 573, L1

  16. [24]

    2019, , 623, A153

    Olofsson , H., Khouri , T., Maercker , M., et al. 2019, , 623, A153

  17. [25]

    S., Scibelli , S., et al

    Sahai , R., Huang , P. S., Scibelli , S., et al. 2022, , 929, 59

  18. [26]

    Schneider , F. R. N., Ohlmann , S. T., Podsiadlowski , P., et al. 2020, , 495, 2796

  19. [27]

    P., Richards , A

    Singh , A. P., Richards , A. M. S., Humphreys , R. M., Decin , L., & Ziurys , L. M. 2023, , 954, L1

  20. [28]

    2019, , 886, 15

    Tokuda , K., Fukui , Y., Harada , R., et al. 2019, , 886, 15

  21. [29]

    S., Figura , C., Cross , J

    Urquhart , J. S., Figura , C., Cross , J. R., et al. 2021, , 500, 3050

  22. [30]

    Wei , D., Schneider , F. R. N., Podsiadlowski , P., et al. 2024, , 688, A87

  23. [31]

    A., Gal-Yam , A., et al

    Yaron , O., Perley , D. A., Gal-Yam , A., et al. 2017, Nature Physics, 13, 510

  24. [32]

    2025, , 697, A167

    Zapartas , E., de Wit , S., Antoniadis , K., et al. 2025, , 697, A167

  25. [33]

    , " * write output.state after.block = add.period write newline

    ENTRY address archiveprefix author booktitle chapter edition editor howpublished institution eprint journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sent...

  26. [34]

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