❈♦♠♠♦♥ ❏❛✈❛❙❝r✐♣t ❉❛t❛ ❚②♣❡s ❖❜❥❡❝ts ✭✇❤✐❝❤ ✐♥❝❧✉❞❡ ❢✉♥❝t✐♦♥s✮✳ ◆✉♠❜❡rs ✭♥♦ ✐♥t❡❣❡rs✮✳ ❆r✐t❤♠❡t✐❝ ❜❛s❡❞ ♦♥ ✻✹✲❜✐t ■❊❊❊✲✼✺✹ st❛♥❞❛r❞✳ ❙tr✐♥❣s✳ ✉♥❞❡❢✐♥❡❞ ❛♥❞ ♥✉❧❧ ✳ ❇♦♦❧❡❛♥s✿ tr✉❡ ❛♥❞ ❢❛❧s❡ ✳ ❖❜❥❡❝ts ❛r❡ ♥♦♥✲♣r✐♠✐t✐✈❡✳ ❆❧❧ ♦t❤❡r t②♣❡s ❛r❡ ♣r✐♠✐t✐✈❡ t②♣❡s✳ ❏❛✈❛❙❝r✐♣t ❇❛s✐❝s
❆ ●❧✐♠♣s❡ ❆t ❖❜❥❡❝ts ❆♥ ♦❜❥❡❝t ✐s ♠❡r❡❧② ❛ ♥❛♠❡❞ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❦❡②✲✈❛❧✉❡ ♣❛✐rs ✭✇❤✐❝❤ ✐♥❝❧✉❞❡ ❢✉♥❝t✐♦♥s✮✳ ❱❛❧✉❡s ❛r❡ r❡❢❡rr❡❞ t♦ ❛s ♦❜❥❡❝t ♣r♦♣❡rt✐❡s ✳ ✴✴♦❜❥❡❝t ❧✐t❡r❛❧ > ① ❂ ④ ❛✿ ✾✱ ❜✿ ✷✱ ⑥ ④ ❛✿ ✾✱ ❜✿ ✷ ⑥ > ①✳❜ ✴✴st❛t✐❝ ❦❡② ♥❛♠❡ ✷ > ①❬✬❛❜❝✬❬✶❪❪ ✴✴❞②♥❛♠✐❝ ❦❡② ♥❛♠❡ ✷ > ❝ ❂ ✬①②✬ ①② > ③ ❂ ④ ❛✿ ✹✷✱ ❝✱ ❜✿ ✽ ⑥ ✴✴❦❡② ❂ ✈❛r✐❛❜❧❡ ♥❛♠❡ ④ ❛✿ ✹✷✱ ❝✿ ✬①②✬✱ ❜✿ ✽ ⑥ > ③ ❂ ④ ❛✿ ✹✷✱ ❬❝❪✿ ✶✾ ⑥ ✴✴❞②♥❛♠✐❝ ❦❡② ④ ❛✿ ✹✷✱ ①②✿ ✶✾ ⑥ ✴✴✈❛r ♥❛♠❡ ❛s ❦❡②✱ ✈❛r ✈❛❧✉❡ ❛s ✈❛❧✉❡ > ④ ❝✱ ③ ⑥ ④ ❝✿ ✬①②✬✱ ③✿ ④ ❛✿ ✹✷✱ ①②✿ ✶✾ ⑥ ⑥ ❏❛✈❛❙❝r✐♣t ❇❛s✐❝s
❆ ●❧✐♠♣s❡ ❆t ❖❜❥❡❝ts ❈♦♥t✐♥✉❡❞✿ ❋✉♥❝t✐♦♥ Pr♦♣❡rt✐❡s ❢✉♥❝t✐♦♥ ✭❛✮ ④ r❡t✉r♥ ❛ ✯ t❤✐s ✳①❀ ⑥ ⑥ ❛ ❂ ④ ①✿ ✷✱ ❢✿ ④ ①✿ ✷✱ ❢✿ ❬❋✉♥❝t✐♦♥✿ ❢❪ ⑥ ✴✴❢✉♥❝t✐♦♥ ♣r♦♣ ✈❛❧✉❡ > ❛✳ ❢ ✭✸✮ ✻ > ❛ ❂ ④ ①✿ ✷✱ ❢ ✭❛✮ ④ r❡t✉r♥ ❛ ✯ t❤✐s ✳①❀ ⑥ ⑥ ④ ①✿ ✷✱ ❢✿ ❬❋✉♥❝t✐♦♥✿ ❢❪ ⑥ > ❛✳ ❢ ✭✸✮ ✻ > ❝ ❂ ✬①②✬ ✬①②✬ > ❛ ❂ ④ ①✿ ✷✱ ❬❝❪✭❛✮ ④ r❡t✉r♥ ❛ ✯ t❤✐s ✳①❀ ⑥ ⑥ ④ ①✿ ✷✱ ①②✿ ❬❋✉♥❝t✐♦♥✿ ①②❪ ⑥ > ❛✳ ①② ✭✹✮ ✽ ❏❛✈❛❙❝r✐♣t ❇❛s✐❝s
◆✉♠❜❡rs ◆♦ ✐♥t❡❣❡rs ✐♥ ❡❛r❧② ❏❙❀ ♣r♦❜❧❡♠❛t✐❝ ❢♦r ✜♥❛♥❝✐❛❧ ❝❛❧❝✉❧❛t✐♦♥s✳ ✩ ♥♦❞❡❥s > ✶✴✵ ■♥❢✐♥✐t② > ✵✴✵ ◆❛◆ > ◆❛◆ ❂❂❂ ◆❛◆ ✴✴■❊❊❊ ❜❡❤❛✈✐♦r❀ ✐♥ ♦t❤❡r ❧❛♥❣✉❛❣❡s t♦♦ ❢❛❧s❡ > ✷✯✯✺✸ ✴✴✯✯ ✐s ❡①♣♦♥❡♥t✐❛t✐♦♥ ♦♣❡r❛t♦r ✾✵✵✼✶✾✾✷✺✹✼✹✵✾✾✷ > ✷✯✯✺✸ ✰ ✶ ✴✴■❊❊❊ ✻✹ ❜✐t ✢♦❛ts ❤❛✈❡ ❛ ✺✸✲❜✐t ♠❛♥t✐ss❛✳ ✾✵✵✼✶✾✾✷✺✹✼✹✵✾✾✷ > ✭✷✯✯✺✸ ✰ ✶✮ ❂❂❂ ✷✯✯✺✸ tr✉❡ > ❇✐❣■♥t ✭✷✯✯✺✸✮ ✰ ✶♥ ✴✴❡①❛❝t ❜✐❣ ✐♥t❡❣❡rs ✾✵✵✼✶✾✾✷✺✹✼✹✵✾✾✸♥ ❏❛✈❛❙❝r✐♣t ❇❛s✐❝s
◆♦r♠❛❧ ❆r✐t❤♠❡t✐❝ ❛♥❞ ❇✐t✇✐s❡ ❖♣❡r❛t♦rs ❯s✉❛❧ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t♦rs ✰ ✭❜♦t❤ ✐♥✜① ❛♥❞ ♣r❡✜①✮✱ ✲ ✭❜♦t❤ ✐♥✜① ❛♥❞ ♣r❡✜①✮✱ ✯ ✱ ✴ ❛♥❞ ✪ ✭r❡♠❛✐♥❞❡r✱ ❤❛s s✐❣♥ ♦❢ ❞✐✈✐❞❡♥❞ ✮✱ ✯✯ ✭♣♦✇❡r✮✳ ❇✐t✇✐s❡ ♦♣❡r❛t♦rs ✫ ✱ ⑤ ✱ ❫ ✱ ⑦ ✱ ❁❁ ✱ ❃❃ ✭❛r✐t❤✮✱ ❃❃❃ ✭❧♦❣✐❝❛❧✮✳ ❇✐t✇✐s❡ ♦♣❡r❛t♦rs ❛❧✇❛②s ✇♦r❦ ✇✐t❤ ✸✷✲❜✐t ✷✬s✲❝♦♠♣❧❡♠❡♥t ✐♥t❡❣❡rs✳ Pr❡✈✐♦✉s ♣r♦♣❡rt② ✉s❡❞ ✐♥ ❛s♠✳❥s t♦ ♦❜t❛✐♥ ❛❝❝❡ss t♦ ♠♦r❡ ❡✣❝✐❡♥t ♠❛❝❤✐♥❡ ✐♥t❡❣❡r ♦♣❡r❛t✐♦♥s✳ ❏❛✈❛❙❝r✐♣t ❇❛s✐❝s
❆r✐t❤♠❡t✐❝ ❖♣❡r❛t♦rs ❊①❛♠♣❧❡s ✴✴✬❴✬ ❛❧❧♦✇❡❞ ❢♦r r❡❛❞❛❜✐❧✐t② > ✶✷✸❴✹✺✻✯✷ ✷✹✻✾✶✷ ✴✴✪ ❤❛s s✐❣♥ ♦❢ ❞✐✈✐❞❡♥❞ > ✲✼✼✪✶✸ ✲✶✷ > ✼✼✪✲✶✸ ✶✷ ✴✴✯✯ ✐s r✐❣❤t ❛ss♦❝✐❛t✐✈❡ > ✷✯✯✷✯✯✸ ✷✺✻ > ✭✷✯✯✷✮✯✯✸ ✻✹ > ✶✽✯✷ ✰ ✼✼✪✶✸ ✴ ✷ ✴✴♦t❤❡r ❜✐♥❛r② ♦♣❡r❛t♦rs ❧❡❢t ❛ss♦❝✐❛t✐✈❡ ✹✷ ❏❛✈❛❙❝r✐♣t ❇❛s✐❝s
❇✐t✇✐s❡ ❖♣❡r❛t♦rs ❊①❛♠♣❧❡s > ✶ ⑤ ✷ ✴✴❜✐t✇✐s❡✲♦r ✸ > ✵①✾✾ ✫ ✵①✸ ✴✴❜✐t✇✐s❡✲❛♥❞❀ ❤❡① ♥♦t❛t✐♦♥ ✶ > ✺ ❫ ✼ ✴✴❜✐t✇✐s❡✲①♦r ✷ > ∼ ✵ ✴✴❜✐t✇✐s❡✲❝♦♠♣❧❡♠❡♥t ✴✴✵①✛✛✛✛ ✐s ✲✶ ✲✶ ✴✴❧❡❢t✲s❤✐❢t > ✸ << ✹ ✴✴① << ♥ ❂❂❂ ① ✯ ✷✯✯♥ ✹✽ ✴✴❛r✐t❤♠❡t✐❝ r✐❣❤t✲s❤✐❢t > ✶✵✵ >> ✸ ✶✷ ✴✴ ① >> ♥ ❂❂❂ ① ✴ ✷✯✯♥ > ❏❛✈❛❙❝r✐♣t ❇❛s✐❝s
▼♦r❡ ♦♥ ❙❤✐❢t ❖♣❡r❛t♦rs ❙❤✐❢t ♦♣❡r❛t♦rs ❝❛♥ ❜❡ ✉s❡❞ t♦ ♠✉❧t✐♣❧② ✭❧❡❢t✲s❤✐❢t✮ ❛♥❞ ❞✐✈✐❞❡ ✭r✐❣❤t✲s❤✐❢t✮ ❜② ♣♦✇❡rs✲♦❢✲✷✳ ❉✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ ❃❃ ✭s✐❣♥✲♣r♦♣❛❣❛t✐♥❣ ♦r ❛r✐t❤♠❡t✐❝ r✐❣❤t✲s❤✐❢t✮ ❛♥❞ ❃❃❃ ✭③❡r♦✲✜❧❧ ♦r ❧♦❣✐❝❛❧ r✐❣❤t✲s❤✐❢t✮✳ ◆♦ ❞✐✛❡r❡♥❝❡ ❢♦r ♥♦♥✲♥❡❣❛t✐✈❡ ♥✉♠❜❡rs✱ ❜✉t ❞✐✛❡r❡♥t r❡s✉❧ts ❢♦r ♥❡❣❛t✐✈❡ ♥✉♠❜❡rs✿ > ✲✾ >> ✶ ✲✺ > ✲✾ >>> ✶ ✷✶✹✼✹✽✸✻✹✸ > ✭✲✾ >>> ✶✮✳ t♦❙tr✐♥❣ ✭✶✻✮ ✬✼❢❢❢❢❢❢❜✬ > ❏❛✈❛❙❝r✐♣t ❇❛s✐❝s
❙tr✐♥❣s ❙tr✐♥❣s ❛r❡ ✐♠♠✉t❛❜❧❡✳ ❈❧❛ss✐❝❛❧❧②✱ str✐♥❣ ❧✐t❡r❛❧s ❛r❡ ❞❡❧✐♠✐t❡❞ ✉s✐♥❣ ❡✐t❤❡r ❞♦✉❜❧❡ q✉♦t❡s ✧ ♦r s✐♥❣❧❡ q✉♦t❡s ✬ ✳ Pr❡❢❡r ✬ ❞❡❧✐♠✐t❡rs s✐♥❝❡ ❡❛s✐❡r t♦ t②♣❡ ♦♥ ♥♦r♠❛❧ ❦❡②❜♦❛r❞s✳ ❇❛❝❦s❧❛s❤❡s ✐♥t❡r♣r❡t❡❞ ❛s ✉s✉❛❧✳ ❈❛♥♥♦t s♣❛♥ ♠✉❧t✐♣❧❡ ❧✐♥❡s✳ ✴✴str✐♥❣ ❝♦♥❝❛t❡♥❛t✐♦♥ > ✬❛✬ ✰ ✬❜✬ ✬❛❜✬ ✴✴✐♥❞❡①✐♥❣✿ r❡s✉❧ts ✐♥ str✐♥❣ ♦❢ ❧❡♥❣t❤ ✶ > ✬❛❜❝✬❬✶❪ ✬❜✬ > ✬❤❡❧❧♦ ✇♦r❧❞✬✳ ✐♥❞❡①❖❢ ✭✬♦✬✮ ✹ > ✬❤❡❧❧♦ ✇♦r❧❞✬✳ ❧❛st■♥❞❡①❖❢ ✭✬♦✬✮ ✼ > ✬❤❡❧❧♦ ✇♦r❧❞✬✳ s✉❜str ✭✸✱ ✹✮ ✴✴❛r❣s✿ ✭st❛rt■♥❞❡①✱ ❧❡♥❣t❤✮ ✴✴tr❡❛t ❛s ❧❡❣❛❝② ❢✉♥❝t✐♦♥ ✬❧♦ ✇✬ ❏❛✈❛❙❝r✐♣t ❇❛s✐❝s
❙tr✐♥❣s ❈♦♥t✐♥✉❡❞ > ✬❤❡❧❧♦ ✇♦r❧❞✬✳ s✉❜str✐♥❣ ✭✸✱ ✹✮ ✴✴❛r❣s✿✭st❛rt■♥❞❡①✱ ❡♥❞■♥❞❡①✮ ✬❧✬ > ✬❤❡❧❧♦ ✇♦r❧❞✬✳ s❧✐❝❡ ✭✻✮ ✬✇♦r❧❞✬ > ✬❤❡❧❧♦ ✇♦r❧❞✬✳ s❧✐❝❡ ✭✶✱ ✹✮ ✴✴❛r❣s✿ ✭st❛rt■♥❞❡①✱ ❡♥❞■♥❞❡①✮ ✬❡❧❧✬ > ✬❤❡❧❧♦ ✇♦r❧❞✬✳ s❧✐❝❡ ✭✲✸✮ ✴✴✐♥❞❡① ❢r♦♠ r✐❣❤t❀ ✲✶ ✐s r✐❣❤t♠♦st ✬r❧❞✬ > ✬❤❡❧❧♦ ✇♦r❧❞✬✳ s❧✐❝❡ ✭✲✸✱ ✲✶✮ ✬r❧✬ ❏❛✈❛❙❝r✐♣t ❇❛s✐❝s
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