Story highlights Israeli Prime Minister Benjamin Netanyahu says Hamas is responsible, "will pay"

Israel has arrested hundreds, demolished homes, closed institutions, he adds

A little-known group supporting the "Islamic State" claims responsibility for teens' deaths

The 3 teens are remembered as "gifted, pure, honest and decent" at their funeral

Mourning three Israeli teenagers at their funeral Tuesday, Israeli Prime Minister Benjamin Netanyahu promised his nation will avenge their deaths at "the hands of evil men" and show its strength in the process.

"Throughout the history of our people, we have proved time and again -- in view of the horrors that we have experienced, the grief and sorrow -- the force of life that pulsates within us overcomes the forces of death," he said in Hebrew, according to a CNN translation.

The Israeli leader lamented the killings of the three "gifted, pure, honest and decent teens," bloodshed that he said shows "a broad moral gulf (that) separates us from our enemies."

"They sanctify death, we sanctify life," Netanyahu said, comparing the teens to those who killed them. "They sanctify cruelty, and we mercy and compassion. That is the secret of our strength."

A short time later -- after the burials of Eyal Yifrach, 19; Gilad Shaar, 16; and Naftali Frankel, a 16-year-old dual U.S.-Israeli citizen, by each other in Modiin -- the Prime Minister spoke again about the three before a security cabinet meeting, saying, "May God avenge their blood."

Photos: Israel mourns 3 dead teenagers Photos: Israel mourns 3 dead teenagers Israel mourns 3 dead teenagers – Mourners of Eyal Yifrach, 19, attend his funeral outside his house in Elad, Israel, on Tuesday, July 1. The bodies of Yifrach and two other Israeli teenagers, Naftali Frankel and Gilad Shaar, were found a day earlier after they were abducted last month on their way home from school in the West Bank. Hide Caption 1 of 6 Photos: Israel mourns 3 dead teenagers Israel mourns 3 dead teenagers – Avi and Rachel Frankel attend the funeral of their son, Naftali, in the West Bank settlement of Nof Ayalon on July 1. He was 16. Hide Caption 2 of 6 Photos: Israel mourns 3 dead teenagers Israel mourns 3 dead teenagers – Friends of Naftali Frankel attend his funeral service. Hide Caption 3 of 6 Photos: Israel mourns 3 dead teenagers Israel mourns 3 dead teenagers – Mourners gather around the body of Gilad Shaar, 16, during his funeral ceremony July 1 in Talmon, West Bank. Hide Caption 4 of 6 Photos: Israel mourns 3 dead teenagers Israel mourns 3 dead teenagers – A young man grieves during Shaar's funeral. Hide Caption 5 of 6 Photos: Israel mourns 3 dead teenagers Israel mourns 3 dead teenagers – People hold Israeli flags during Shaar's funeral procession in Hashmonaim, West Bank. Hide Caption 6 of 6

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"Whoever was involved in the kidnapping and the murder will bear the consequences," Netanyahu said Tuesday evening. "We will neither rest nor slacken until we reach the last of them. And It does not matter where they will try to hide."

These actions are well underway. The Israeli leader noted that, already, hundreds of Hamas activists have been arrested, dozens of institutions in Gaza have closed and homes have been demolished.

This is along with stepped-up airstrikes in Gaza -- Palestinian land controlled by Hamas, a militant Islamic organization that denied it is behind the abductions.

The Israeli military said it launched 34 strikes targeting terror infrastructure targets since Sunday evening in response to 18 rockets from Gaza.

The Palestinian Cabinet on Tuesday claimed 12 Palestinians have been killed since the military operation to find the teens began June 13; it did not identify them by name, nor detail how or when they died.

CNN reported earlier, citing Palestinian medical sources, that at least one man died from a heart attack during a raid on his West Bank home last month; at least five were killed in clashes with the Israeli military; and two men accused by the Israeli military of being involved in recent rocket attacks were killed in an airstrike last week in Gaza.

Addressing "all of those who are ... happy at our sadness," Israeli President Shimon Peres warned at Tuesday's funeral, "Terror is a boomerang."

"I know that we will eventually get our hands on the murderers," Peres said, "and they will be punished."

Israeli leader: 'Hamas will pay'

The bodies of Yifrach, Shaar and Frankel were found Monday, nearly three weeks after their abduction while hitchhiking home from school in the West Bank.

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Palestinian Authority President Mahmoud Abbas, who heads the Fatah party and is based in the West Bank, has condemned the abductions, even as the Palestinian Cabinet has criticized what it calls "Israel's illegal measures" in response.

Yet Mark Negev, a spokesman for Netanyahu, suggested Monday that Abbas' Palestinian Authority bears some responsibility for what happened because "the terrorists came from areas under Palestinian Authority control and returned to territories under Palestinian Authority control."

Still, while Abbas and the Palestinian Authority are clearly feeling the pressure, it's nothing compared to that facing Hamas Just listen to Netanyahu on Tuesday evening: "Hamas is responsible. Hamas will pay, and Hamas will continue to pay."

The Israel Defense Forces named two men -- Marwan Kawasme and Omar Abu Aysha -- as the "Hamas terrorists who kidnapped" the boys. Their West Bank homes were destroyed overnight.

Palestinian news agency Ma'an reported Tuesday that a little-known group called Ansar as-Dawla al-Islamiya (Supporters of the Islamic State) had claimed responsibility for the killings and threatened to "slaughter" the Palestinian Authority. CNN could not independently verify the claim.

Hamas has made no such claim. Still, as it bears the brunt of the retaliation, the group warned that if Netanyahu "brings a war on Gaza, the gates of hell will open to him."

Obama: Israel, Palestinian Authority should work together

The teens' kidnappings and killings -- as well as the Israeli response -- have spurred strong reactions well outside the Middle East, including among U.S. politicians, New York's mayor and the Vatican.

The advocacy group Human Rights Watch, for instance, said the killings "would amount to a war crime if committed by an armed group" even as it urged Israel to avoid "collective punishment."

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British Prime Minister David Cameron has called the teens' deaths "an appalling and inexcusable act of terror."

U.S. President Barack Obama similarly condemned what he called a "senseless act of terror against innocent youth." He reiterated "our full support to Israel and the Palestinian Authority to find the perpetrators of this crime and bring them to justice."

"And I encourage Israel and the Palestinian Authority to continue working together in that effort," Obama said. "I also urge all parties to refrain from steps that could further destabilize the situation."