Fighters linked to the Islamic State in Iraq and the Levant (ISIL) have beheaded a Frenchman abducted in Algeria in a video posted online, but France's president has pledged that France would not give in to blackmail.
Francois Hollande said the "cruel and cowardly" murder of Herve Gourdel, 55, would only strengthen France's resolve to pursue its air war against ISIL in Iraq.
Gourdel was kidnapped on Sunday by Jund al-Khilafa, or Soldiers of the Caliphate, while hiking in a national park that was once a magnet for tourists but later became a sanctuary for armed groups.
His beheading followed France's rejection of the group's 24-hour ultimatum to halt its air raids in Iraq - part of a US-led campaign against IS.
Hollande, whose government was the first to join the US administration in carrying out air strikes against the self-declared jihadists, pledged that France would stay the course.
"France is going through an ordeal through the murder of one of its citizens, but France will never give in to blackmail," he told the UN General Assembly.
"The fight against terrorism must continue and be stepped up."
US President Barack Obama voiced solidarity with France over the killing, which came after two American journalists and a British aid worker were beheaded by ISIL in similar videotaped executions.
Chairing a special UN Security Council meeting, Obama looked at his French counterpart and said: "We stand with you and the French people as you grieve this terrible loss and as you stand up against terror in defence of liberty."
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The video of Gourdel's killing begins with a clip of Hollande at a press conference where he announced strikes against ISIL in Iraq.
It then shows Gourdel on his knees with his hands behind his back in front of four fighters, their faces covered.
Gourdel briefly expresses his love for his family before one of the abductors reads a speech in which he denounces the actions of the "French criminal crusaders" against Muslims in Algeria, Mali and Iraq.
He says the beheading is to "avenge the victims in Algeria ... and support the caliphate" proclaimed by ISIL in Iraq and Syria.
The US has built a coalition of more than 50 nations to fight ISIL, after the rebels seized large parts of Syria and Iraq and committed widespread atrocities, including beheadings and crucifixions.
France has ruled out joining military operations in Syria, where a US-led coalition began strikes against ISIL on Tuesday.
In a speech on Wednesday that kicked off a parliamentary session about France's engagement in Iraq, Manuel Valls, France prime minister, said it was not the cause of Gourdel's abduction.
"It's not our intervention that is exposing us to terrorism. The threat has been there for a long time. And that's why we are acting," Valls said.
The Algerian government called it "hateful" and expressed its "determination to continue the struggle against terrorism in all its forms and guarantee the protection and security of foreign nationals".
Algiers also pledged to "continue to mobilise all forces possible to find the killers so that they can be punished, and to find Herve Gourdel's body," Hollande said after speaking to Abdelmalek Sellal, Algerian prime minister.
Gourdel had only arrived in Algeria on Saturday and was seized the following day while hiking in the heart of the Djurdjura National Park, an area with dense forests, deep gorges and picturesque lakes.
The mountains became a hideout for fighters in the 1990s who later swore allegiance to al-Qaeda, and security forces have been unable to dislodge them.
The group that abducted Gourdel was formed recently after splintering from Al-Qaeda in the Islamic Maghreb (AQIM), which experts say has raked in some $120m in ransom payments in the past eight years.