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* Brazil's real reverses early gains * Petrobras boosts Bovespa as oil rises * Mexican peso steady after 3-day losing run Dec 14 (Reuters) - Brazil's real strengthened to close to 5 per dollar on Monday before reversing course, while the Mexican peso steadied after a three-day losing run driven by optimism around the rollout of COVID-19 vaccines. The real fell 0.8% to 5.0872 per dollar, having earlier firmed to as much as 5.0094 in an attempt to breach the key 5-per-dollar level last hit in mid-June. The reversal of early gains came as central bank data showed Brazilian economic activity grew in October, although the pace of recovery from the coronavirus pandemic lost momentum. However, against a broadly weakening dollar, the Mexican , Colombian and Chilean pesos all firmed. Investors placed riskier bets on hopes of a speedy economic recovery as the first inoculations in the United States with the Pfizer/BioNTech COVID-19 vaccine outside trial were set to be administered. Meanwhile, the Brazilian government unveiled its long-awaited national vaccination plan over the weekend, with an initial goal of vaccinating about one-fourth of the population in the first half of 2021. The country's benchmark stock index rose 0.2%, with shares in state-run oil firm Petrobras SA gaining 0.8% after crude prices climbed on hopes of improving fuel demand. The company also said it had kicked off the non-binding phase of the sale of a 50% stake in the Marlim field in the Campos basin. Brazilian telecom firms Telefonica Brasil SA and TIM SA added 1% and 2.8% respectively, as the companies were expected to win the auction for Oi SA's cellular operations in a joint bid with America Movil's Claro. Key Latin American stock indexes and currencies: Latest Daily % change MSCI Emerging Markets 1252.17 -0.44 MSCI LatAm 2401.25 -0.42 Brazil Bovespa 115304.84 0.15 Mexico IPC 43362.19 -0.57 Chile IPSA 4098.80 0.32 Argentina MerVal 54428.20 1.125 Colombia COLCAP 1386.56 0.65 Currencies Latest Daily % change Brazil real 5.0872 -0.80 Mexico peso 20.1365 -0.01 Chile peso 731.4 0.14 Colombia peso 3425.19 0.30 Peru sol 3.5857 0.17 Argentina peso (interbank) 82.4500 -0.28 (Reporting by Shashank Nayar in Bengaluru; Editing by Kirsten Donovan)