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(PͰ࠾༻͞Εͨ 1BUUFSOEFGFBUJOH2VJDLTPSUͷ঺հ )JSPNV/BLBNVSB !QPSJO 1BUUFSOEFGFBUJOH2VJDLTPSUJO(P

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தଜ߂෢ ΤϜεϦʔͷ"*ɾػցֶशνʔϜͷιϑτ΢ΣΞ ΤϯδχΞͰओʹݕࡧج൫पΓΛ୲౰͍ͯ͠Δɻ ֤छΞΧ΢ϯτQPSJOͰ΍ͬͯ·͢ɻ )*SPNV/BLBNVSB The Gopher was designed by Renée French

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*OUSPEVDUJPO

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ϦϦʔεϊʔυݟΔͱTPSUύοέʔδͷ಺෦ΞϧΰϦζϜ͕Կ΍Βมߋ͞ΕͨͬΆ͍ (PͷTPSUύοέʔδͷมߋ

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࣮ࡍʹTPSU[TPSUGVODHPͱTPSU[TPSUJOUFSGBDFHPʹͦΕͧΕQERTPSU@GVODɺQERTPSUͱ͍͏ ؔ਺͕௥Ճ͞Ε͍ͯΔɻݺͼग़͍ͯ͠Δͷ͸TPSU4PSUͱTPSU4MJDF (PͷTPSUύοέʔδͷมߋ

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ࠓճ͸1BUUFSOEFGFBUJOH2VJDLTPSU<> ͷ࿦จΛಡΜͰ͍͖ɺ(PͰͲͷΑ͏ʹ࣮ ૷͞Ε͍ͯΔ͔؆୯ʹݟ͍ͯ͘ 1BUUFSOEFGFBUJOH2VJDLTPSU [1] https://arxiv.org/abs/2106.05123

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1BUUFSOEFGFBUJOH2VJDLTPSUͷͬ͘͟Γͱͨ͠ཧղ ূ໌ͳͲࡉ͔͍࿩͸εΩοϓͯ͠ͳΜͱͳ͘ԿΛ΍ͬͯΔͷ͔ͷཧղΛ໨ࢦ͢ ͜ͷൃදͰಘΕΔ΋ͷ

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1BUUFSOEFGFBUJOH2VJDLTPSU

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*OUSPTPSUͷ෮श 1BUUFSOEFGFBUJOH2VJDLTPSU͸Πϯτϩιʔτͷվྑ൛ͳͷͰɺ؆୯ʹΠϯτϩιʔτΛ෮ श͓͖ͯ͠·͢ɻ Πϯτϩιʔτ͸ۙ೥࠷΋࢖༻͞Ε͍ͯΔϋΠϒϦουιʔτΞϧΰϦζϜͰ͋Γɺૠೖιʔ τɺώʔϓιʔτɺΫΠοΫιʔτΛ૊Έ߹ΘͤͨϋΠϒϦουΞϧΰϦζϜͰ͢ɻ ώϡʔϦεςΟοΫͳํ๏Ͱɺ͍ͭͲͷઓུΛ࠾༻͢Δ͔Λܾఆ͠·͢ɻྫ͑͹ɺ࠶ؼͷਂ͞ ͕ਂ͘ͳΓ͗͢Δͱώʔϓιʔτʹ੾Γସ͑ͨΓɺύʔςΟγϣϯͷαΠζ͕খ͘͞ͳΓ͗͢ Δͱૠೖιʔτʹ੾Γସ͑ͨΓ͠·͢ɻ

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1BUUFSOEFGFBUJOH2VJDLTPSU͸ΠϯτϩιʔτΛ֦ு͓Αͼվྑͨ͠΋ͷͰ͢ɻ ޙͰৄ͘͠આ໌͍͖ͯ͠·͕͢ɺ1BUUFSOEFGFBUJOH2VJDLTPSUͷ؊͸؆୯ʹݴ͏ͱ ύʔςΟγϣϯॲཧΛ͍͍ײ͡ʹͨ͠ ࠷ѱέʔεΛ্ख͘ݕ஌ͯ͠ϑΥʔϧόοΫ͢ΔΑ͏ʹͨ͠ ͷͭͰ͢ɻ 1BUUFSOEFGFBUJOH2VJDLTPSU֓ཁ

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ॏෳΛؚΉ໿ཁૉͷγʔέϯεΛιʔτ͢Δ1BUUFSOEFGFBUJOH2VJDLTPSUͷࢹ֮Խɻ 1BUUFSOEFGFBUJOH2VJDLTPSU֓ཁ

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1BSUJUJPOJOH

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1BUUFSOEFGFBUJOH2VJDLTPSU͸Φϥϯμࠃض໰୊ͷߟ͑ํ͕ϕʔεʹͳ͍ͬͯΔͷͰɺ·ͣ͸Φϥϯμ ࠃض໰୊Λ෮श͠·͢ɻ Φϥϯμࠃض໰୊͸ɺɺͷΈΛؚΉγʔέϯεΛಉ͡ཁૉͰύʔςΟγϣϯ͢Δ໰୊Ͱ͢ɻྫ͑͹Լ هͷJOΛPVUʹม׵͢Δ໰୊Ͱ͢ɻ JO PVU 1BSUJUJPOJOH

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1BSUJUJPOJOH ΞϧΰϦζϜ͸୯७Ͱઢܗ࣌ؒͰղ͚·͢ɻNJEͱMPXΛ഑ྻͷ࠷ॳͷཁૉͷJOEFYʹηοτ͠ɺIJHIΛ࠷ ޙͷཁૉͷJOEFYʹηοτ͠·͢ɻ w ΋͠NJEͳΒMPXͱNJEͷཁૉΛSFQMBDFͯ͠IJHIͰσΫϦϝϯτ w ΋͠NJEͳΒNJEΛΠϯΫϦϝϯτ w ΋͠NJEͳΒIJHIͱNJEͷཁૉΛSFQMBDFͯ͠IJHIΛσΫϦϝϯτ

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MPX NJE IJHI

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MPX NJE IJHI MPX NJE IJHI NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI

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MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI NJEͳͷͰɺNJE

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MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI NJEͳͷͰɺNJE NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX

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MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI NJEͳͷͰɺNJE NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX

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MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI NJEͳͷͰɺNJE NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX NJEͳͷͰɺNJE

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MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI MPX NJE IJHI NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI NJEͳͷͰɺNJE NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX NJEͳͷͰɺMPXͱSFQMBDFͯ͠NJE MPX NJEͳͷͰɺNJE NJEͳͷͰɺIJHIͱSFQMBDFͯ͠IJHI NJE͕IJHIΛ௒͑ͨͷͰऴྃ

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1BSUJUJPOJOHJO1BUUFSOEFGFBUJOH2VJDLTPSU ୯७ͳΫΠοΫιʔτͷ࣮૷Ͱ͸ɺ౳͍͠ൺֱཁૉΛಉ͡ύʔςΟγϣϯʹ഑ஔ͢Δ͜ͱʹ ΑΓϺ O? ͷ࠷ѱͷέʔε͕Ҿ͖ى͜͞Ε·͢ɻ ౳͍͠ཁૉΛޮ཰తʹॲཧ͢Δʹ͸ɺ౳͍͠ཁૉΛूΊͯɺͦΕΑΓେ͖͍ཁૉɺখ͍͞ཁ ૉͷ෼ׂʹͰ͖Ε͹ɺͦͷޙͷιʔτͰ౳͍͠ཁૉΛؚΉύʔςΟγϣϯͷιʔτΛε ΩοϓͰ͖·͢ɻ·͞ʹΦϥϯμࠃض໰୊ͱಉ౳ͷࡾ෼ׂͷߟ͑ํ͕࢖͑ͦ͏Ͱ͢ɻ ͜ͷΑ͏ʹύʔςΟγϣϯΛ෼ׂͰ͖Δͱخ͍͠

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1BSUJUJPOJOHJO1BUUFSOEFGFBUJOH2VJDLTPSU ͜Ε·Ͱʹઆ໌ͨ͠Φϥϯμͷࠃض໰୊ͷղ๏Λ୯७ʹద༻͢Δͱɺ͢΂ͯͷཁૉΛεϫο ϓ͢ΔલʹϐϘοτͱ౳͍͔͠Ͳ͏͔Λ໌ࣔతʹνΣοΫ͢Δඞཁ͕͋Γ·͢ɻ͜Ε͸ɺ౳ ͍͠ཁૉ͕ଟ਺ଘࡏ͢Δ͔Ͳ͏͔ʹؔ܎ͳ͘ൃੜ͠ɺฏۉతͳέʔεͰ͸ύϑΥʔϚϯε͕ ௿Լ͠·͢ɻ ͜ͷ࿦จͰ͸͜ͷ໰୊ʹରॲ͢Δ৽͍͠ղܾࡦΛٞ࿦͓ͯ͠ΓɺLݸͷҟͳΔཁૉΛ࣋ͭೖ ྗʹରͯ͠0 OL ͷ࠷ѱέʔεʹऩ·Δख๏ΛఏҊ͍ͯ͠·͢ɻ1BUUFSOEFGFBUJOH 2VJDLTPSUͰ͸͜ͷख๏ΛϕʔεʹΠϯτϩιʔτΛվྑ͍ͯ͠·͢ɻ

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1BSUJUJPO-FGU3JHIU 1 1 QBSUJUJPO@SJHIU QBSUJUJPO@MFGU 1BUUFSOEFGFBUJOH2VJDLTPSU͸ͭͷύʔςΟγϣϯ༻ؔ਺Λ૊Έ߹ΘͤͯઌఔͷύʔςΟγϣϯ Λୡ੒͠·͢ɻͭ͸ϐϘοτQʹ౳͍͠ཁૉΛࠨͷύʔςΟγϣϯʹάϧʔϓԽ QBSUJUJPOMFGU ͠ɺ΋͏ͭ͸ϐϘοτʹ౳͍͠ཁૉΛӈͷύʔςΟγϣϯ QBSUJUJPOSJHIU ʹάϧʔϓԽ͠·͢ɻ ͜ͷύʔςΟγϣϯͷ૊Έํͷར఺͸ʮཁૉ͕ϐϘοτͷ஋Ҏ্͔Ͳ͏͔ʯͱ͍͏৚݅ҰݸͰ൑ఆͰ ͖Δ͜ͱͰ͢ɻ

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1BUJUJPOJO(P (PͰ͸TPSUύοέʔδ಺ͷQBUJUJPOؔ਺͕ύʔςΟγϣϯΛ୲͏ɻ࣮ࡍʹؔ਺ΛFYQPSUͯ͠Ұճͩ ͚ಈ͔ͯ͠ΈΔͱ͔֬ʹQBUJUJPO@MFGUͷ݁ՌΛಘΒΕ͍ͯΔɻ

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1BSUJUJPO-FGU3JHIU QBSUJUJPOؔ਺Λར༻ͯ͠࠶ؼతʹγʔέϯεΛύʔςΟγϣϯ͍͖ͯ͠·͢ɻ ྫ͑͹ԼهਤͷΑ͏ʹɺϐϘοτQͰЋΛύʔςΟγϣϯ͠ɺ࠶ؼతʹӈଆͷύʔςΟγϣϯΛϐ ϘοτRͱͯ͠ЌΛύʔςΟγϣϯ͍͖ͯ͠·͢ɻ Q͕Rʹ౳͘͠ͳ͍৔߹͸ɺ্هͷεςοϓΛ࠶ؼతʹద༻͠·͢ɻ࠷ऴతʹQRͱͳͬͨ࣌͸ɺಉ ͡஋ΛਅΜதʹूΊͨߏ੒ʹͰ͖͍ͯ·͢Լਤɻ ͪͳΈʹɺ͋Δ఺RͰЌ͕ۭʹͳͬͨ৔߹ɺQʹ౳͍͠ཁૉ͸ͳ͘ɺЋΛ࠷ॳʹ෼ׂͨ͠ͱ͖ʹ ෼ׂ͕ߦΘΕͨͱ݁࿦෇͚Δ͜ͱ͕Ͱ͖·͢ɻ

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https://www.youtube.com/watch?v=jz-PBiWwNjc ͔ΒҾ༻

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1JWPU4FMFDUJPO

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+PIO5VLFZ`TOJOUIFS ϐϘοτͷબ୒ʹ͓͍ͯɺෆۉߧύʔςΟγϣϯॲཧΛආ͚ΔͨΊʹɺͳΔ΂͘தԝ஋Λબ୒͢Δ͜ ͱ͕ཧ૝Ͱ͢ɻҰํͰେ͖ͳೖྗͷ৔߹͸ద੾ͳϐϘοτΛબ୒͢Δͷ͸ύϑΥʔϚϯεͷ໰୊͕ൃ ੜ͠·͢ɻͦ͜Ͱ͜ͷ࿦จͰ͸༗໊ͳ+PIO5VLFZ`TOJOUIFSͱ͍͏ख๏ͰϐϘοτΛબ୒͠·͢ɻ ݫີͳதԝ஋Λܭࢉ͢Δʹ͸͢΂ͯͷσʔλϙΠϯτΛอଘ͠ɺͦΕΒΛฒ΂ସ͑ͯதؒ஋Λબ୒͢ Δඞཁ͕͋Γ·͕͢ɺ͜ͷํ๏Ͱ͋Ε͹ૉૣ͘େࡶ೺ͳதԝ஋ͷۙࣅ͕ಘΒΕ·͢ɻ

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+PIO5VLFZ`TOJOUIFSJO(P ࣮ࡍͷ(Pͷ࣮૷Ͱ͸ϐϘοτΛબ୒͢Δαϒγʔέϯεͷ௕͞ʹΑͬͯϐϘοτબ୒ํ๏Λ੾Γସ ͑·͢ɻ௕͕͞ҎԼͳΒ୯७ʹதԝ஋ΛͱΓɺҎԼͳΒ୯७ʹ࠷ॳɺதԝɺ࠷ޙͷͭͷ஋ͷத ԝ஋ΛͱΓɺҎ্ͳΒ+PIO5VLFZ`TOJOUIFSΛར༻͠·͢ɻ

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#BE1BUJUJPO

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#BE1BUJUJPO 1BUUFSOEFGFBUJOH2VJDLTPSU͸ɺࢦఆͨ͠ϐϘοτύʔηϯλΠϧQ ೚ҙͷ஋ ΑΓ΋ෆۉߧͳ෼ׂ ૢ࡞ΛෆྑύʔςΟγϣϯͱݺͼ·͢ ͪͳΈʹϐϘοτύʔηϯλΠϧQ͕ͩͱཧ૝త͸ύʔ ςΟγϣϯʹͳΓ·͢ ɻྫ͑͹ԼهͷΑ͏ͳγʔέϯεͰϐϘοτύʔηϯλΠϧͷϙΠϯτ ࠓճ͸ Λ࠾༻ͯ͠͠·͏ͱ΄ͱΜͲύʔςΟγϣϯ͕ػೳ͠·ͤΜɻ

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#BE1BUJUJPO :VWBM'JMNVT<>͸Qͷ࠷దͳέʔεͱൺֱͯ͠ΫΠοΫιʔτͷ଎౓௿ԼΛௐ΂·ͨ͠ɻ γϟϊϯͷ஋ΤϯτϩϐʔΛར༻ͯ͠ɺԼهͷΑ͏ͳάϥϑ͕ॻ͚·͢ɻ ͜ͷάϥϑ͔Βɺྫ͑͹Qͷ৔߹͚ͩͷ৔߹͸ɺཧ૝తͳ৔߹ͱൺ΂͚ͯͩ஗͘ͳΔ͜ͱ ͕Θ͔Γ·͢ɻQΛԼճΔͱύϑΥʔϚϯεͷ௿Լ͕ܹ͘͠ͳΔ͜ͱ΋Θ͔Γ·͢ɻ͜ͷͨΊɺ Qͷᮢ஋Λઃఆ͠ɺͦΕΑΓԼճΔෆྑύʔςΟγϣϯͷέʔε͕ଟ͚Ε͹ϑΥʔϧόοΫ͢Δઓུ ΛऔΓ·͢ɻ [3] https://cs.stackexchange.com/questions/31928/solving-recurrence-relation-with-two-recursive-calls/31930#31930

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#BE1BUJUJPO ෆྑύʔςΟγϣϯΛෳ਺ճૺ۰͢ΔͷΛճආ͢ΔͨΊʹԼهεςοϓΛߦ͍·͢ɻ w ࠷ॳʹɺΧ΢ϯλʔͱͯ͠MPH αϒγʔέϯε௕ ʹઃఆ͠·͢ w ෆྑύʔςΟγϣϯʹૺ۰͢ΔͨͼʹɺΧ΢ϯλʔΛσΫϦϝϯτ͠·͢ w ࠶ؼݺͼग़͠ͷ։࢝࣌ʹΧ΢ϯλʔ͕ͷ৔߹ɺΫΠοΫιʔτͰ͸ͳ͘ɺώʔϓιʔτΛ࢖༻ ࿦จͰ͸͜ͷϐϘοτύʔηϯλΠϧQͷᮢ஋Λͱ͓ͯ͠Γ(P΋ʹͳ͍ͬͯ·͢ɻ ͜ͷεΩʔϜͷར఺͸ɺΞʔΩςΫνϟ͕มߋ͞Εͨ৔߹ ώʔϓιʔτͷ୅ΘΓʹผͷ࠷ѱͷέʔε ͷιʔτΞϧΰϦζϜΛ࢖༻͍ͯ͠Δ৔߹ͳͲ ʹɺͦΕʹ߹ΘͤͯQΛௐ੔Ͱ͖Δ͜ͱͰ͢ (Pͷ৔ ߹͸͕ϋʔυίʔυ͞Ε͍ͯͯมߋ͸Ͱ͖ͳ͍ ɻ

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#BE1BUJUJPO (PͰ͸QERTPSUؔ਺ͷMJNJUҾ਺͕Χ΢ϯλʔ ʹ͋ͨΓɺ͜Ε͕ʹͳΔͱώʔϓιʔτʹ੾ ΓସΘΓ·͢ɻ

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#BE1BUJUJPO ͪͳΈʹͪΐͬͱͨ͠޻෉ͱͯ͠Χ΢ϯλʔ͸γʔέϯεͷ௕͞ͷॴཁϏοτ਺ͱͯؒ͠ ઀తʹMPH@ O Λܭࢉ͍ͯ͠Δɻ

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1BUUFSOEFGFBUJOH2VJDLTPSUJO(P

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1BUUFSOEFGFBUJOH2VJDLTPSUJO(P (Pʹ࣮૷͞Εͨ1BUUFSOEFGFBUJOH2VJDLTPSU͸࿦จͱൺ΂ͯԼهͷΑ͏ͳมߋ͕Ճ͑ΒΕͯ ͍·͢ɻ w #MPDL2VJDLTPSUΛ࢖ͬͨ࠷దԽΛແޮ কདྷ௥Ճ͞ΕΔՄೳੑ͸͋Γͦ͏ w ϕϯνϚʔΫ݁Ռʹैͬͯɺ͍͔ͭ͘ͷύϥϝʔλʔΛมߋ

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(Pʹ͓͚ΔϕϯνϚʔΫ͸ JTTVFͰఏग़͞Ε͍ͯ·͢ɻ΄ͱ ΜͲͷέʔεͰύϑΥʔϚϯε͕վળ͠ ͍ͯ·͢ɻ

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$PODMVTJPO

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$PODMVTJPO (Pʹ࣮૷͞Εͨ1BUUFSOEFGFBUJOH2VJDLTPSU͸ϕϯνϚʔΫ্༷ʑͳύλʔϯͰύϑΥʔϚ ϯε͕վળ͍ͯ͠Δɻ 1BUUFSOEFGFBUJOH2VJDLTPSU͸ओʹύʔςΟγϣϯॲཧ΍ϐϘοτબ୒ɺෆྑύʔςΟγϣϯ ݕ஌Ͱɺ2VJDLTPSUͷ࠷ѱέʔεΛճආͰ͖ΔΑ͏ʹ޻෉͞Ε͍ͯΔɻ (PʹΞοϓάϨʔυͯ͠TPSU͕ͲΕ͘Β͍ૣ͘ͳ͔ͬͨੋඇࢼͯ͠Έ͍ͯͩ͘͞ɻΑΓࡉ ͔͍޻෉΍ɺৄ͍͠ূ໌ͳͲ͸࿦จΛࢀর͍ͩ͘͞ɻ

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