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"I will recommend that we review what training we do, awareness training, for Members of Congress and their staffs in terms of reasonable security measures to be taken at their offices and when they're in public events," the Republican told a news outlet in his home state of California. "And I would recommend very strongly that Members of Congress revisit the relationship they have with local law enforcement, and sit down and talk with local law enforcement about what's appropriate in terms of a presence of law enforcement when a Member of Congress has a public gathering of some sort."
Rep. Debbie Wasserman Schultz, who is a good friend of Giffords, said on NBC's "Meet the Press" Sunday that there "needs to be a wake-up call for lawmakers who treat security in a cavalier" way. The Democrat said she always has local police at events in her Florida district to protect her and her constituents.
Several Members took to newspapers and the airwaves to express views about what could be done to boost their own safety in light of the rare attempted murder of one of their colleagues.
Among them was Assistant Leader James Clyburn (D-S.C.), who suggested on "Fox News Sunday" that instead of cutting House operating budgets by five percent, leaders should add money to Members' Representational Allowances to beef up security.
But Crenshaw assured the budget cut resolution passed last week will not affect Member safety.
"That five percent cut to leadership and office budgets should have no impact on security issues," he said. "I don't know that people spend money from their office accounts for security."
The last time a Member of Congress was assassinated was in 1978, when Rep. Leo Ryan (D-Calif.) was shot and killed leaving Jonestown, Guyana, after meeting with members of Jim Jones' cult.
Democratic Caucus Chairman John Larson (Conn.) organized a call with House Sergeant-At-Arms Bill Livingood to brief Members on the events in Arizona and security precautions.
"We have a number of new Members who know nothing about this, this is their first experience," he said after visiting with Giffords' staff on Sunday.
Rep. Tom Cole spent the bulk of last month interviewing House officials, including Livingood and Capitol Police Chief Phillip Morse, to make recommendations about how to modify House operations during the GOP's transition to power.
"Obviously we will carefully review this incident and see what we can learn," the Oklahoma Republican said. "And we will fully examine any recommendations pertaining to the security of Members, staff and constituents made by either the Capitol Police or the Sergeant-at-Arms."
Members of leadership always have a Capitol Police security detail but other Members only get one when a heightened threat necessitates it. There was no indication that Giffords was in danger, so she did not have added security, Gainer said.
That underscores the reality that there's no surefire way to protect all 535 Members all the time. "I don't think that should paralyze us," Gainer said. "If you balance what happened in Tucson against the tens of thousands of appearances that Members make across this country, more things are going right than are going wrong."
Jessica Brady contributed to this report.