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The US had publicly announced plans to send Apollo 8 to orbit the Moon in December. The earliest we could send a mission was to launch on December 21, 1968. The launch window was dictated by the need to get to the Moon when it was at the same phase for the planned landing attempts in 1969, and also to time the mission so that splashdown back on Earth would be in 'friendly' waters.
We did not know what the Russian plans were, but we could guess. The Soviet launch window was earlier in the month, and there was a real fear that the Russians would send men to the Moon before us. Indeed, the Russians had sent up circumlunar spacecraft in September and November that year. It wasn't publicly known in the US at the time, but these two spacecraft were not just Moon probes - they could have carried a manned crew. Had the USSR been willing to take the risk, they could have beaten us to the Moon by at least a month. They could not have made a landing in 1969, but their having sent men to the Moon before the US would certainly have lessened our feeling of accomplishment.
As it turned out, the decision not to send men on the November flight was the correct one. The 'crew' of biological specimens perished when a faulty O-ring gasket caused the cabin to depressurize before reentry, and a parachute deployed early, causing the capsule to crash. Either failure would have killed a human crew. The Soviets were not able to fix the design faults in time to make an early December launch, which enabled the US to be the first to send men around the Moon.