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OTL NASA did develop plans for Mars and Venus flybys using Apollo hardware, scheduled to be launched in the mid to late 1970s.
So, perhaps instead of pledging to land a man on the moon and return him safely to Earth by the end of the decade, Kennedy pledges to send astronausts on a flyby mission to Mars by, say, America's bicentenniel.
Is such a program plausible, and, if so, could these eaarly flybys lead to further manned exploration of Mars?