West Palm Beach, Florida - Florida's governor said Tuesday the state will run its own investigation and seek tougher charges against the man suspected of trying to shoot Donald Trump, potentially including attempted murder, as he described federal gun violation counts as inadequate.

Governor Ron DeSantis also said he was assigning the probe of the assassination plot to the office of Florida's attorney general, Ashley Moody, in part because of his distrust in federal agencies' ability to remain neutral in cases related to Trump.

"In my judgment, it's not in the best interest of our state or our nation to have the same federal agencies that are seeking to prosecute Donald Trump leading this investigation," DeSantis told a press conference.

Ryan Routh was detained after fleeing Trump's club in West Palm Beach, where the former president was playing golf on Sunday when Secret Service agents discovered a rifle barrel sticking out from a treeline and opened fire on the suspect.

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Routh (58) was charged with possession of a firearm as a convicted felon and possession of a firearm with an obliterated serial number at his initial court appearance.

DeSantis said Florida was going further.

"We have jurisdiction under state law to pursue an investigation and prosecution under the most serious, readily provable offense, which would be attempted murder, which would carry a maximum sentence of life in prison," he said.

"If we do anything but go to the fullest extent of the law on this, then I think we're sending a signal that maybe this isn't that bad. And that's unacceptable."

US Attorney General Merrick Garland meanwhile said the entire Justice Department, including the FBI and US Attorney's Office for the Southern District of Florida, which is prosecuting the Routh case, are "coordinating closely" with local and state law enforcement partners.

While Florida's attorney general Moody stressed that running a parallel investigation "doesn't mean it's a turf war" with federal authorities, she echoed DeSantis's impartiality concerns.

Moody, standing at DeSantis's side, also said Florida will investigate how things went "terribly wrong."

"Someone was allowed to remain on the periphery of a golf course in a treeline for 12 hours and get within 500 feet" of the former president, she said.