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Assuming the Easter Islanders 'kept their precessional clock running', their Old Year would be made to stop not with December 31 but with November 14 (i.e. 4 days after the Gregorian π day), when the Tau-ono stars were close to the Full Moon:

Nov 14 15 16 (320) 17 18
May 15 16 (136) 17 18 19
Ca9-10 Ca9-11 Ca9-12 Ca9-13 Ca9-14 (242)
kua rere ki te marama e moa haati kava e moa
π Cor. Borealis, Cor Serpentis (238.1), Chow (238.6) κ Serpentis (239.3), δ Cor. Borealis, Tiānrǔ (239.5), χ Lupi, (239.6), ω Serpentis (239.7), Ba, χ Herculis (239.8). κ Cor. Borealis, ρ Serpentis (239.9) β Tr. Austr. (240.3), κ Tr. Austr. (240.4), ρ Scorpii (240.8) Iklīl al Jabhah-15 / Anuradha-17 / Room-4  υ Herculis (242.3), ρ Cor. Borealis (242.4), ι Cor. Borealis (242.5), ξ Scorpii (242.7)

Schedir

ξ Lupi, λ Cor. Borealis (241.1), Zheng (241.2), VRISCHIKA (241.3), ε Cor. Borealis (241.5),  Dschubba (241.7), η Lupi (241.9)
Al Thurayya-27 / Krittikā-3 / Hairy Head-18 Menkhib (57.6)

Porrima

Zaurak (58.9) λ Tauri (59.3), ν Tauri (59.9)
Atiks, Rana (55.1), CELAENO, ELECTRA, TAYGETA (55.3), MAIA, ASTEROPE, MEROPE (55.6)  ALCYONE (56.1), PLEIONE, ATLAS (56.3)
TAU-ONO

Then could follow for instance a kava ceremony (Ca9-13), maybe with 'cocks' crying out to make the new Year come awake. Not until perhaps 3 weeks later would the new Sun King be ready, though:

Nov 19 20 (324) 21 22 23 24
May 20 (140) 21 22 23 24 25
Ca9-15 Ca9-16 Ca9-17 Ca9-18 Ca9-19 Ca9-20
i te mauga pu hia E rima ki te henua koia ku honui erua maitaki ko koe ra
16h (243.5) ψ Scorpii (244.6), Lesath (244.8) χ Scorpii (245.1), Yed Prior, δ Tr. Austr. (245.5) Yed Posterior, Rukbalgethi Shemali (246.6). δ Apodis (246.7), ο Scorpii (246.8) Heart-5 ρ Ophiuchi (248.1), Kajam (248.3), χ Ophiuchi (248.5), She Low, ι Tr. Austr. (248.7), ζ Tr. Austr. (248.8)
Acrab, Jabhat al Akrab (243.3), θ Lupi, Rutilicus (243.5), Marfik (243.7), φ Herculis (243.8) σ SCORPII (247.0), Hejian (247.2), ψ Ophiuchi (247.7)
4h (60.9) no star listed (61) Beid (62.2)

Vindemiatrix

Al Dabarān-2 Hyadum II (64.2) Net-19
no star listed (60)

Cor Caroli

HYADUM I (63.4) θ² Tauri, AIN, θ¹ Tauri (65.7)
Nov 25 26 (330) 27 28 29 30
May 26 27 28 29 30 (150) 31
Ca9-21 Ca9-22 Ca9-23 Ca9-24 Ca9-25 Ca9-26
ka mau - i te inoino ka iri ka hua i te inoino te hau tea te inoino kua iri kua puo te inoino
Al Kalb-16 / Jyeshtha-18

ANA-MURI

γ Apodis (250.1), σ Herculis (250.3), θ Tr. Austr. (250.6), τ Scorpii (250.7) Han (251.0) ζ Herculis, η Tr. Austr. (252.1), η Herculis, β Apodis (252.5) Atria (253.9) Tail-6
ANTARES (249.1), Marfik, φ Ophiuchi (249.5),  ω Ophiuchi (249.8) Wei, η Arae (254.3), DENEBAKRAB (254.7)
no star listed (66) no star listed (67) Rohini-4

ANA-MUA

no star listed (69) no star listed (70) no star listed (71)
ALDEBARAN (68.2), Theemin (68.5)
December 1 (335)
June 1 (152)
Ca9-27 (255)
etoru gagata hakaariki kia raua
ι Ophiuchi (255.3), Grafias (255.4)
no star listed (72)
December 2 (336) 3 4
June 2 (153) 3 (520) 4
Ca10-1 (256) Ca10-2 Ca10-3
Erua inoino kua hua te vai
κ Ophiuchi (256.2), ζ Arae (256.5), ε Arae (256.8), Cujam (256.9) no star listed (257) 17h (258.7)
no star listed (258)
Hassaleh (73.6) Almaaz (74.7), Haedus I (74.8) Haedus II (75.9)
December 5 6  7 8 (342)
June 5 6 (157) 7 8 (525)
Ca10-4 Ca10-5 (260) Ca10-6 Ca10-7
te kiore - te inoino kua oho te rima kua kai - ihe nuku hoi Tupu te toromiro kua noho te vai
Mula-19 Nodus I (260.0), π Herculis (260.7), Ras Algethi (260.8) Sarin (261.0), ο Ophiuchi (261.4)

Alrisha

ξ Ophiuchi (262.2), θ Ophiuchi, ν Serpentis, ζ, ι Apodis (262.4), ι Arae (262.8), ρ Herculis (262.9)
Sabik (259.7), η SCORPII (259.9)
5h (76.1) μ Leporis (77.6) ĸ Leporis (78.0), Rigel (78.1), Capella (78.4), ο Columbae (78.8)

Thuban

λ Leporis (79.6)

Arcturus

ε Leporis (76.0), Cursa (76.4), λ Eridani (76.7)

From 320 (November 16) - at the opposite side of May 17 (137) - to Rigel and Capella rising with the new Sun in December 7 and with Arcturus culminating at midnight in the following day number 342 there were 22 days.

This may have been when on Easter Island the 'Toromiro' would grow in the 'water', when the first 260 days from March 21 were over. In December 8 according to the Gregorian calendar, but in day 342 - 27 = 315 (= 314 + 1) according to the calendar which was kept alive.

Evidently we have to use this updated and running calendar also on side a.