What did Western leaders think would happen after the USSR's satellite states adopted communism?

The Cold War was a geopolitical chess game between the United States and Western Europe from Jha against the Soviets each alliance sought to gain more power in different parts of the world Especially that the Soviet Union has acquired adjacent areas from Europe and the United States such as the crisis in Cuba, Vietnam and the Berlin Wall

we cannot forget for understand this game (chess) who was exactly, that The Eastern Bloc was the group of Communist states of Central and Eastern Europe, East Asia, and Southeast Asia under the hegemony of the USSR during the Cold War (1947-91),

and opposition to the non-communist Western Bloc