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ACROSS

 1 Objectively, I pester an Indian tribe: It supplies three letters for the
wordplay in each remaining Across entry (9)

 6 Politely refuse medical professional (5)

 9 Allocates some atoms (7)

10 They’re liable to run away from poets, losing face (7)

11 Military assignment! (6,7)

13 Put together most of logo (8)

14 Excoriate that woman… (6)

18 …ear head of security for European capital (6)

20 Animated film failing in front of an entire continent (8)

23 Madam’s IRA invested in obsolescent technology (7,6)

26 Heavy drinkers with drink dispensers (7)

27 Finishes off blueprint for Veracruz wine bar that’s often swinging (7)

28 Mother’s doctrine (5)

29 Supreme Court ran out of order (9)

DOWN

 1 Evangelist’s target (4)

 2 Observes broken section (7)

 3 Space (small and medium) taken up by general participants in a
wedding (9)

 4 Roguish scar disfigured friend (8)

 5 Prime minister spots upcoming bivouac (6)

 6 Begins lifting tight bundle on poles (5)

 7 Red streetcar carrying half a dozen uphill (7)

 8 Disturbing siren gets you out of bed (5)

12 Backing up a little information on the computer, perhaps (4)

15 What you’d find in a cell: blasting cap, mostly (9)

16 Cheese is manufactured upside down (4)

17 One making a keen escape with log (8)

19 Start off playing pool or craps and laughing derisively (7)

21 Outlandish crackpot as ruler (7)

22 Category that includes part of purse! (6)

23 Located, like, in bed (5)

24 Vietnamese leader: “I get it, a commotion” (3-2)

25 Sell the fifth object (4)

SOLUTION TO PUZZLE NO. 3462

ACROSS 1 CAM (rev.) + BRID(G)E 6 SE[n]DER 9 RE(GAT)TA (tear anag.) 10 B(ELI)EVE 11 C + LOVER LEAVES 14 TEN E.T.’S 15 DE(SIGN)ER 17 RE(SIGN)ED 19 G + OLDEN 21 pun 25 “why, key key” 26 COLLI(D)E 27 anag. 28 THOU + SANDS

DOWN 1 C(ARI + CAT)URE 2 MYG (rev.) + OODNESS (anag.) 3 R(ETRE)AT 4 DR + AWL 5 EM + B + RACED 6 SA(LIE)RI 7 rev. 8 rev. 12 IN(AD + DITIO)N (idiot anag.) 13 “Gran’s tress” spoonerism 16 BEN(EDIC)T (dice anag.) 18 G + UNFIRE 20 O(RIO)LES 22 WAC + KO 23 T + WIG 24 anag.