Santorum vows to soldier on after 3rd place finish in S.C.

Walking into chants of "we pick Rick" from an audience at the military academy The Citadel, Rick Santorum vowed to continue his journey to the Republican nomination after finishing third in South Carolina Saturday. Santorum politely congratulated his opponents, saying "Newt kicked butt," referring to the winner in the Palmetto state's primary, former House Speaker Newt Gingrich.

In an interview with CNN before those remarks, Santorum was less charitable toward Gingrich.

"Those who worked with congressman Gingrich know what's in store if he's the nominee of our party," he said, referring to his rival's reputation as an erratic leader.

Santorum sought to put the best spin on his third place finish in South Carolina, where his message of social conservatism is more welcome than it was in New Hampshire.

"Three states, three winners. What a great country," Santorum said, reminding his supporters and those watching on television that he received 34 more votes than Romney in the non-binding Iowa caucuses January 3. Mitt Romney won New Hampshire January 10.

But Santorum faces a difficult road ahead. The former Pennsylvania senator trails both Romney and Gingrich in national polls.

"It's game on again. We're going to have it go on for a long time," Santorum told CNN.

Santorum said he was headed Sunday to Florida, which holds its primary January 31.

"Then we will go to Arizona and Colorado and everywhere else," he told his supporters. Nevada then Maine are the next contests after Florida.

While talking to supporters after his speech, Santorum was doused with glitter as protestors repeatedly yelled "bigot."

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Ron Paul gears up for the long haul

Full CBS News coverage: Rick Santorum