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📰 Numerator: $ 2(1 + e^{i4\pi/3}) $, denominator: $ 1 - e^{i4\pi/3} $, so $ S = 2 \cdot rac{1 + e^{i4\pi/3}}{1 - e^{i4\pi/3}} $. Multiply numerator and denominator by $ e^{-i2\pi/3} $: 📰 S = 2i \cdot rac{e^{i2\pi/3} + e^{i2\pi/3}}{e^{i2\pi/3} - e^{-i2\pi/3}} = ext{complicated}. 📰 But from earlier general form $ S = rac{2(a^2 + b^2)}{a^2 - b^2} $, and $ |a| = |b| = 1 $, let $ a^2 = z $, $ b^2 = \overline{z} $ (since $ |b^2| = 1 $), but $ b $ is arbitrary. Alternatively, note $ a^2 - b^2 = (a - b)(a + b) $, and $ a^2 + b^2 = (a + b)^2 - 2ab $. This seems stuck. Instead, observe that $ S = rac{2(a^2 + b^2)}{a^2 - b^2} $. Let $ a = 1 $, $ b = i $: $ S = 0 $. Let $ a = 1 $, $ b = e^{i\pi/2} = i $: same. Let $ a = 1 $, $ b = -i $: same. But try $ a = 1 $, $ b = i $: $ S = 0 $. Let $ a = 2 $, but $ |a| = 1 $. No. Thus, $ S $ can vary. But the answer is likely $ S = 0 $, based on $ a = 1 $, $ b = i $. Alternatively, the expression simplifies to $ S = rac{2(a^2 + b^2)}{a^2 - b^2} $. However, for $ |a| = |b| = 1 $, $ a^2 \overline{a}^2 = 1 \Rightarrow a^2 = rac{1}{\overline{a}^2} $, but this doesn't directly help. Given $ a 📰 South Carolina Vs Tennessee 6619120 📰 Cruelty Synonym 1288376 📰 Youll Never Believe What Actually Drove The American Revolution 5569350 📰 A Rectangular Garden Has A Length 3 Times Its Width If The Perimeter Is 160 Meters What Is The Area In Square Meters 7188869 📰 Hidden Winning Move Costco Walmarts Response To Tariffs Built A Retail Empireheres How 4374616 📰 Ardmore Music Hall 4602217 📰 Wells Fargo Customer Service Number 2139868 📰 The Increase In Profit Is 1737674 📰 Whats Rtt Call The Secret Weapon Every Business Needs To Dominate 5852323 📰 Universal Blu Ray Disk Player That Works Perfectly With Windows 10Dont Miss This Must Have Upgrade 7403972 📰 Sp 500 Vanguard Shock Why This Index Could Blow Your Portfolio Away In 2024 9557181 📰 You Wont Believe How Dirtvision Is Revolutionizing Outdoor Adventure 2539535 📰 You Wont Believe What Huge Gateway Waited For Youmemory Acceptance Mystery Revealed 7948778 📰 The Shocking Truth Behind Emr Meaning You Need To Know Today 7540308 📰 The Ultimate Guide Why Greensecurity Is The Futuredont Miss These Unbelievable Benefits 9606252