In the OTL, the USSR only took American efforts to land a man on the moon seriously and undertook efforts for a Manned Lunar program seriously in 1964. By then, it looked too late, and the Americans landed a man on the moon in 1969 while the USSR was still working on their N-1 in the early 1970s, before the program was canceled for lack of success and lack of perceived need given NASA's success in landing on the moon.
But what if the Soviet Union took US plans to land on the moon seriously before 1964, and therefore seriously undertook an effort to land on the moon earlier than they did?