How did the third would countries play into the plans of the two would superpower during the Cold War

proxy wars

Explanation:

third world countries had unstable governments, leading to US and Russian interferrence through direct (military takeover) or indirect (aid) means in order to contain either communism or capitalism.

If you recall your World War II history, the United States and the Soviet Union were allies during the war. Together, these nations and others brought about the downfall of Hitler's Third Reich. When the war ended in 1945, these two countries emerged as world super powers.