WASHINGTON (Reuters) - U.S. National Security Adviser Susan Rice said on Monday that the United States cannot allow countries like North Korea or Iran think that Washington would not react to a chemical weapons attack in Syria.

"We cannot allow terrorists bent on destruction, or a nuclear North Korea, or an aspiring nuclear Iran, to believe for one minute that we are shying away from our determination to back up our longstanding warnings," Rice said in her first major speech since taking over as President Barack Obama's top security adviser in July.

In her speech at the New America Foundation think tank, Rice said the United States intends to renew its push for the U.N.-sponsored Geneva peace process in Syria following any limited military strikes that are currently being considered by the U.S. Congress.

(Reporting by Susan Cornwell and Deborah Charles)